FE ( Fundamentals of Engineering ) Supplied-Reference Handbook 8th Edition Revised

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Eighth Edition REVISED

This document may be downloaded from the NCEES Web site, but it may not be copied, reproduced, or posted online without the express written permission of the National Council of Examiners for Engineering and Surveying® (NCEES®).

FUNDAMENTALS OF ENGINEERING SUPPLIED-REFERENCE HANDBOOK National Council of Examiners for Engineering and Surveying

Published by the National Council of Examiners for Engineering and Surveying® 280 Seneca Creek Road, Clemson, SC 29631 800-250-3196 www.ncees.org ©2008 by the National Council of Examiners for Engineering and Surveying®. All rights reserved. ISBN 978-1-932613-37-7 Printed in the United States of America October 2008 Revised

PREFACE The Fundamentals of Engineering (FE) examination was developed by the National Council of Examiners for Engineering and Surveying (NCEES) as the first step toward professional engineering licensure. It is designed for students completing a bachelor degree program in engineering. The FE exam consists of two 4-hour sessions—one administered in the morning and the other in the afternoon. The morning session tests the subject matter covered by the first 90 semester credit hours of engineering coursework, while the afternoon session tests upper-division subject knowledge covering the remainder of required degree coursework. The FE Supplied-Reference Handbook is the only reference material allowed during the examination. A copy of this handbook will be made available to each examinee during the exam and returned to the proctor before leaving the exam room at the conclusion of the administration. Examinees are prohibited from bringing any reference materials with them to the exam. As well, examinees are prohibited from writing in the Reference Handbook during the exam administration. There are no sample questions or solutions included in the Reference Handbook—examinees can self-test using one of the NCEES FE Sample Questions and Solutions books, CD-ROMs, or online practice exams, all of which may be purchased by calling (800) 250-3196 or visiting our Web site at www.ncees.org. The material included in the FE Supplied-Reference Handbook is not all-encompassing or exhaustive. NCEES in no event shall be liable for not providing reference material to support all the questions in the FE exam. Some of the basic theories, conversions, formulas, and definitions examinees are expected to know have not been included. Furthermore, the FE Supplied-Reference Handbook may not contain some special material required for the solution of a particular exam question—in such a situation, this material will be included in the question itself. In the interest of constant improvement, NCEES reserves the right to revise and update the FE Supplied-Reference Handbook as it deems appropriate. Each FE exam will be administered using the latest version of the FE SuppliedReference Handbook. To report suspected errata in this book, please e-mail your correction using our online feedback form. Examinees are not penalized for any errors in the Reference Handbook that affect an exam question. For current exam specifications, a list of approved calculators, study materials, errata, guidelines for special accommodations requests, and other information about the exams and the licensure process, visit www.ncees.org or call (800) 250-3196.

iii

TABLE OF CONTENTS Exam Specifications ......................................................................................................1 Units ............................................................................................................................19 Conversion Factors .....................................................................................................20 Mathematics ................................................................................................................21 Mechanics of Materials ...............................................................................................33 Engineering Probability and Statistics ........................................................................40 Statics ..........................................................................................................................49 Dynamics ....................................................................................................................54 Fluid Mechanics ..........................................................................................................62 Thermodynamics .........................................................................................................73 Heat Transfer ...............................................................................................................84 Transport Phenomena .................................................................................................90 Biology ........................................................................................................................91 Chemistry ..................................................................................................................100 Materials Science/Structure of Matter ......................................................................104 Computer Spreadsheets .............................................................................................109 Measurement and Controls .......................................................................................110 Engineering Economics ............................................................................................114 Ethics .........................................................................................................................121 Chemical Engineering ...............................................................................................123 Civil Engineering ......................................................................................................134 Environmental Engineering ......................................................................................170 Electrical and Computer Engineering .......................................................................193 Industrial Engineering ...............................................................................................215 Mechanical Engineering ...........................................................................................231 Index .........................................................................................................................249

v

EXAM SPECIFICATIONS Fundamentals of Engineering (FE) Examination Effective October 2005 •

The FE examination is an 8-hour supplied-reference examination: 120 questions in the 4-hour morning session and 60 questions in the 4-hour afternoon session.

•

The afternoon session is administered in the following seven modules—Chemical, Civil, Electrical, Environmental, Industrial, Mechanical, and Other/General engineering.

•

Examinees work all questions in the morning session and all questions in the afternoon module they have chosen.

MORNING SESSION (120 questions in 12 topic areas) Approximate Percentage of Test Content

Topic Area I.

Mathematics A. Analytic geometry B. Integral calculus C. Matrix operations D. Roots of equations E. Vector analysis F. Differential equations G. Differential calculus

15%

II.

Engineering Probability and Statistics A. Measures of central tendencies and dispersions (e.g., mean, mode, standard deviation) B. Probability distributions (e.g., discrete, continuous, normal, binomial) C. Conditional probabilities D. Estimation (e.g., point, confidence intervals) for a single mean E. Regression and curve fitting F. Expected value (weighted average) in decision-making G. Hypothesis testing

7%

III.

Chemistry A. Nomenclature B. Oxidation and reduction C. Periodic table D. States of matter E. Acids and bases F. Equations (e.g., stoichiometry) G. Equilibrium H. Metals and nonmetals

9%

IV.

Computers A. Terminology (e.g., memory types, CPU, baud rates, Internet) B. Spreadsheets (e.g., addresses, interpretation, “what if,” copying formulas) C. Structured programming (e.g., assignment statements, loops and branches, function calls)

7%

V.

Ethics and Business Practices A. Code of ethics (professional and technical societies) B. Agreements and contracts C. Ethical versus legal D. Professional liability E. Public protection issues (e.g., licensing boards)

7%

EXAM SPECIFICATIONS—MORNING

1

VI.

Engineering Economics A. Discounted cash flow (e.g., equivalence, PW, equivalent annual FW, rate of return) B. Cost (e.g., incremental, average, sunk, estimating) C. Analyses (e.g., breakeven, benefit-cost) D. Uncertainty (e.g., expected value and risk)

VII.

Engineering Mechanics (Statics and Dynamics) A. Resultants of force systems B. Centroid of area C. Concurrent force systems D. Equilibrium of rigid bodies E. Frames and trusses F. Area moments of inertia G. Linear motion (e.g., force, mass, acceleration, momentum) H. Angular motion (e.g., torque, inertia, acceleration, momentum) I. Friction J. Mass moments of inertia K. Impulse and momentum applied to: 1. particles 2. rigid bodies L. Work, energy, and power as applied to: 1. particles 2. rigid bodies

VIII.

Strength of Materials A. Shear and moment diagrams B. Stress types (e.g., normal, shear, bending, torsion) C. Stress strain caused by: 1. axial loads 2. bending loads 3. torsion 4. shear D. Deformations (e.g., axial, bending, torsion) E. Combined stresses F. Columns G. Indeterminant analysis H. Plastic versus elastic deformation

7%

IX.

Material Properties A. Properties 1. chemical 2. electrical 3. mechanical 4. physical B. Corrosion mechanisms and control C. Materials 1. engineered materials 2. ferrous metals 3. nonferrous metals

7%

X.

Fluid Mechanics A. Flow measurement B. Fluid properties C. Fluid statics D. Energy, impulse, and momentum equations E. Pipe and other internal flow

7%

2


8%

10%

XI.

Electricity and Magnetism A. Charge, energy, current, voltage, power B. Work done in moving a charge in an electric field (relationship between voltage and work) C. Force between charges D. Current and voltage laws (Kirchhoff, Ohm) E. Equivalent circuits (series, parallel) F. Capacitance and inductance G. Reactance and impedance, susceptance and admittance H. AC circuits I. Basic complex algebra

9%

XII.

Thermodynamics A. Thermodynamic laws (e.g., 1st Law, 2nd Law) B. Energy, heat, and work C. Availability and reversibility D. Cycles E. Ideal gases F. Mixture of gases G. Phase changes H. Heat transfer I. Properties of: 1. enthalpy 2. entropy

7%


3

AFTERNOON SESSION IN CHEMICAL ENGINEERING (60 questions in 11 topic areas)

Topic Area

Approximate Percentage of Test Content

I.

Chemistry A. Inorganic chemistry (e.g., molarity, normality, molality, acids, bases, redox, valence, solubility product, pH, pK, electrochemistry) B. Organic chemistry (e.g., nomenclature, structure, qualitative and quantitative analyses, balanced equations, reactions, synthesis)

10%

II.

Material/Energy Balances A. Mass balance B. Energy balance C. Control boundary concept (e.g., black box concept) D. Steady-state process E Unsteady-state process F. Recycle process G. Bypass process H. Combustion

15%

III.

Chemical Engineering Thermodynamics A. Thermodynamic laws (e.g., 1st Law, 2nd Law) B. Thermodynamic properties (e.g., internal thermal energy, enthalpy, entropy, free energy) C. Thermodynamic processes (e.g., isothermal, adiabatic, isentropic) D. Property and phase diagrams (e.g., T-s, h-P, x-y, T-x-y) E. Equations of state (e.g., van der Waals, Soave-Redlich-Kwong) F. Steam tables G. Phase equilibrium and phase change H. Chemical equilibrium I. Heats of reaction J. Cyclic processes and efficiency (e.g., power, refrigeration, heat pump) K. Heats of mixing

10%

IV.

Fluid Dynamics A. Bernoulli equation and mechanical energy balance B. Hydrostatic pressure C. Dimensionless numbers (e.g., Reynolds number) D. Laminar and turbulent flow E. Velocity head F. Friction losses (e.g., pipe, valves, fittings) G. Pipe networks H. Compressible and incompressible flow I. Flow measurement (e.g., orifices, Venturi meters) J. Pumps, turbines, and compressors K. Non-Newtonian flow L. Flow through packed beds

10%

4

EXAM SPECIFICATIONS—CHEMICAL

V.

Heat Transfer A. Conductive heat transfer B. Convective heat transfer C. Radiation heat transfer D. Heat transfer coefficients E. Heat exchanger types (e.g., plate and frame, spiral) F. Flow configuration (e.g., cocurrent/countercurrent) G. Log mean temperature difference (LMTD) and NTU H. Fouling I. Shell-and-tube heat exchanger design (e.g., area, number of passes)

10%

VI.

Mass Transfer A. Diffusion (e.g., Fick’s 1st and 2nd laws) B. Mass transfer coefficient C. Equilibrium stage method (efficiency) D. Graphical methods (e.g., McCabe-Thiele) E. Differential method (e.g., NTU, HETP, HTU, NTP) F. Separation systems (e.g., distillation, absorption, extraction, membrane processes) G. Humidification and drying

10%

VII.

Chemical Reaction Engineering A. Reaction rates and order B. Rate constant (e.g., Arrhenius function) C. Conversion, yield, and selectivity D. Series and parallel reactions E. Forward and reverse reactions F. Energy/material balance around a reactor G. Reactions with volume change H. Reactor types (e.g., plug flow, batch, semi-batch, CSTR) I. Homogeneous and heterogeneous reactions J. Catalysis

10%

VIII.

Process Design and Economic Optimization A. Process flow diagrams (PFD) B. Piping and instrumentation diagrams (P&ID) C. Scale-up D. Comparison of economic alternatives (e.g., net present value, discounted cash flow, rate of return) E. Cost estimation

10%

IX.

Computer Usage in Chemical Engineering A. Numerical methods and concepts (e.g., convergence, tolerance) B. Spreadsheets for chemical engineering calculations C. Statistical data analysis

5%

X.

Process Control A. Sensors and control valves (e.g., temperature, pressure) B. Dynamics (e.g., time constants, 2nd order, underdamped) C. Feedback and feedforward control D. Proportional, integral, and derivative (PID) controller concepts E. Cascade control F. Control loop design (e.g., matching measured and manipulated variables) G. Tuning PID controllers and stability (e.g., Method of Ziegler-Nichols, Routh Test) H. Open-loop and closed-loop transfer functions

5%


5

XI.

6

Safety, Health, and Environmental A. Hazardous properties of materials (e.g., corrosive, flammable, toxic), including MSDS B. Industrial hygiene (e.g., noise, PPE, ergonomics) C. Process hazard analysis (e.g., using fault-tree analysis or event tree) D. Overpressure and underpressure protection (e.g., relief, redundant control, intrinsically safe) E. Storage and handling (e.g., inerting, spill containment) F. Waste minimization G. Waste treatment (e.g., air, water, solids)


5%

AFTERNOON SESSION IN CIVIL ENGINEERING (60 questions in 9 topic areas)

Topic Area


I.

Surveying A. Angles, distances, and trigonometry B. Area computations C. Closure D. Coordinate systems (e.g., GPS, state plane) E. Curves (vertical and horizontal) F. Earthwork and volume computations G. Leveling (e.g., differential, elevations, percent grades)

11%

II.

Hydraulics and Hydrologic Systems A. Basic hydrology (e.g., infiltration, rainfall, runoff, detention, flood flows, watersheds) B. Basic hydraulics (e.g., Manning equation, Bernoulli theorem, open-channel flow, pipe flow) C. Pumping systems (water and wastewater) D. Municipal water distribution systems E. Reservoirs (e.g., dams, routing, spillways) F. Groundwater (e.g., flow, wells, drawdown) G. Sewer collection systems (storm and sanitary)

12%

III.

Soil Mechanics and Foundations A. Index properties and soil classifications B. Phase relations (air-water-solid) C. Laboratory and field tests D. Effective stress (buoyancy) E. Retaining walls (e.g., active pressure/passive pressure) F. Shear strength G. Bearing capacity (cohesive and noncohesive) H. Foundation types (e.g., spread footings, piles, wall footings, mats) I. Consolidation and differential settlement J. Seepage K. Slope stability (e.g., fills, embankments, cuts, dams) L. Soil stabilization (e.g., chemical additives, geosynthetics)

15%

IV.

Environmental Engineering A. Water quality (ground and surface) B. Air quality C. Solid/hazardous waste D. Sanitary sewer system loads E. Basic tests (e.g., water, wastewater, air) F. Environmental regulations G. Water treatment and wastewater treatment (e.g., primary, secondary, tertiary)

12%

EXAM SPECIFICATIONS—CIVIL

7

V.

Transportation A. Streets and highways 1. geometric design 2. pavement design 3. intersection design B. Traffic analysis and control 1. safety 2. capacity 3. traffic flow 4. traffic control devices

12%

VI.

Structural Analysis A. Force analysis of statically determinant beams, trusses and frames B. Deflection analysis of statically determinant beams, trusses and frames C. Stability analysis of beams, trusses and frames D. Column analysis (e.g., buckling, boundary conditions) E. Loads and load paths (e.g., dead, live, moving) F. Elementary statically indeterminate structures

10%

VII.

Structural Design A. Codes (e.g., AISC, ACI, NDS, AISI) B. Design procedures for steel components (e.g., beams, columns, beam-columns, tension members, connections) C. Design procedures for concrete components (e.g., beams, slabs, columns, walls, footings)

10%

VIII.

Construction Management A. Procurement methods (e.g., design-build, design-bid-build, qualifications based) B. Allocation of resources (e.g., labor, equipment, materials, money, time) C. Contracts/contract law D. Project scheduling (e.g., CPM, PERT) E. Engineering economics F. Project management (e.g., owner/contractor/client relations, safety) G. Construction estimating

10%

IX.

Materials A. Concrete mix design B. Asphalt mix design C. Test methods (e.g., steel, concrete, aggregates, asphalt) D. Properties of aggregates E. Engineering properties of metals

8

EXAM SPECIFICATIONS—CIVIL

8%

AFTERNOON SESSION IN ELECTRICAL ENGINEERING (60 questions in 9 topic areas)

Topic Area


I.

Circuits A. KCL, KVL B. Series/parallel equivalent circuits C. Node and loop analysis D. Thevenin/Norton theorems E. Impedance F. Transfer functions G. Frequency/transient response H. Resonance I. Laplace transforms J. 2-port theory K. Filters (simple passive)

16%

II.

Power A. 3-phase B. Transmission lines C. Voltage regulation D. Delta and wye E. Phasors F. Motors G. Power electronics H. Power factor (pf) I. Transformers

13%

III.

Electromagnetics A. Electrostatics/magnetostatics (e.g., measurement of spatial relationships, vector analysis) B. Wave propagation C. Transmission lines (high frequency)

7%

IV.

Control Systems A. Block diagrams (feed forward, feedback) B. Bode plots C. Controller performance (gain, PID), steady-state errors D. Root locus E. Stability

10%

V.

Communications A. Basic modulation/demodulation concepts (e.g., AM, FM, PCM) B. Fourier transforms/Fourier series C. Sampling theorem D. Computer networks, including OSI model E. Multiplexing

9%

EXAM SPECIFICATIONS—ELECTRICAL

9

VI.

Signal Processing A. Analog/digital conversion B. Convolution (continuous and discrete) C. Difference equations D. Z-transforms

8%

VII.

Electronics A. Solid-state fundamentals (tunneling, diffusion/drift current, energy bands, doping bands, p-n theory) B. Bias circuits C. Differential amplifiers D. Discrete devices (diodes, transistors, BJT, CMOS) and models and their performance E. Operational amplifiers F. Filters (active) G. Instrumentation (measurements, data acquisition, transducers)

15%

VIII.

Digital Systems A. Numbering systems B. Data path/control system design C. Boolean logic D. Counters E. Flip-flops F. Programmable logic devices and gate arrays G. Logic gates and circuits H. Logic minimization (SOP, POS, Karnaugh maps) I. State tables/diagrams J. Timing diagrams

12%

IX.

Computer Systems A. Architecture (e.g., pipelining, cache memory) B. Interfacing C. Microprocessors D. Memory technology and systems E. Software design methods (structured, top-down bottom-up, object-oriented design) F. Software implementation (structured programming, algorithms, data structures)

10%

10

EXAM SPECIFICATIONS—ELECTRICAL

AFTERNOON SESSION IN ENVIRONMENTAL ENGINEERING (60 questions in 5 topic areas)

Topic Area


I.

Water Resources A. Water distribution and wastewater collection B. Water resources planning C. Hydrology and watershed processes D. Fluid mechanics and hydraulics

25%

II.

Water and Wastewater Engineering A. Water and wastewater B. Environmental microbiology/ecology C. Environmental chemistry

30%

III.

Air Quality Engineering A. Air quality standards and control technologies B. Atmospheric sciences

15%

IV.

Solid and Hazardous Waste Engineering A. Solid waste engineering B. Hazardous waste engineering C. Site remediation D. Geohydrology E. Geotechnology

15%

V.

Environmental Science and Management A. Industrial and occupational health and safety B. Radiological health and safety C. Radioactive waste management D. Environmental monitoring and sampling E. Pollutant fate and transport (air/water/soil) F. Pollution prevention and waste minimization G. Environmental management systems

15%

EXAM SPECIFICATIONS—ENVIRONMENTAL

11

AFTERNOON SESSION IN INDUSTRIAL ENGINEERING (60 questions in 8 topic areas)

Topic Area


I.

Engineering Economics A. Discounted cash flows (equivalence, PW, EAC, FW, IRR, loan amortization) B. Types and breakdown of costs (e.g., fixed, variable, direct and indirect labor, material, capitalized) C. Analyses (e.g., benefit-cost, breakeven, minimum cost, overhead, risk, incremental, life cycle) D. Accounting (financial statements and overhead cost allocation) E. Cost estimating F. Depreciation and taxes G. Capital budgeting

15%

II.

Probability and Statistics A. Combinatorics (e.g., combinations, permutations) B. Probability distributions (e.g., normal, binomial, empirical) C. Conditional probabilities D. Sampling distributions, sample sizes, and statistics (e.g., central tendency, dispersion) E. Estimation (point estimates, confidence intervals) F. Hypothesis testing G. Regression (linear, multiple) H. System reliability (single components, parallel and series systems) I. Design of experiments (e.g., ANOVA, factorial designs)

15%

III.

Modeling and Computation A. Algorithm and logic development (e.g., flow charts, pseudo-code) B. Spreadsheets C. Databases (e.g., types, information content, relational) D. Decision theory (e.g., uncertainty, risk, utility, decision trees) E. Optimization modeling (decision variables, objective functions, and constraints) F. Linear programming (e.g., formulation, primal, dual, graphical solution) G. Math programming (network, integer, dynamic, transportation, assignment) H. Stochastic models (e.g., queuing, Markov, reliability) I. Simulation (e.g., event, process, Monte Carlo sampling, random number generation, steady-state vs. transient)

12%

IV.

Industrial Management A. Principles (e.g., planning, organizing) and tools of management (e.g., MBO, re-engineering) B. Organizational structure (e.g., functional, matrix, line/staff) C. Motivation theories (e.g., Maslow, Theory X, Theory Y) D. Job evaluation and compensation E. Project management (scheduling, PERT, CPM)

10%

12

EXAM SPECIFICATIONS—INDUSTRIAL

V.

Manufacturing and Production Systems A. Manufacturing systems (e.g., cellular, group technology, flexible, lean) B. Process design (e.g., number of machines/people, equipment selection, and line balancing) C. Inventory analysis (e.g., EOQ, safety stock) D. Forecasting E. Scheduling (e.g., sequencing, cycle time, material control) F. Aggregate planning (e.g., JIT, MRP, MRPII, ERP) G. Concurrent engineering and design for manufacturing H. Automation concepts (e.g., robotics, CIM) I. Economics (e.g., profits and costs under various demand rates, machine selection)

13%

VI.

Facilities and Logistics A. Flow measurements and analysis (e.g., from/to charts, flow planning) B. Layouts (e.g., types, distance metrics, planning, evaluation) C. Location analysis (e.g., single facility location, multiple facility location, storage location within a facility) D. Process capacity analysis (e.g., number of machines/people, trade-offs) E. Material handling capacity analysis (storage & transport) F. Supply chain design (e.g., warehousing, transportation, inventories)

12%

VII.

Human Factors, Productivity, Ergonomics, and Work Design A. Methods analysis (e.g., improvement, charting) and task analysis (e.g., MTM, MOST) B. Time study (e.g., time standards, allowances) C. Workstation design D. Work sampling E. Learning curves F. Productivity measures G. Risk factor identification, safety, toxicology, material safety data sheets (MSDS) H. Environmental stress assessment (e.g., noise, vibrations, heat, computer-related) I. Design of tasks, tools, displays, controls, user interfaces, etc. J. Anthropometry, biomechanics, and lifting

12%

VIII.

Quality A. Total quality management theory (e.g., Deming, Juran) and application B. Management and planning tools (e.g., fishbone, Pareto, quality function deployment, scatter diagrams) C. Control charts D. Process capability and specifications E. Sampling plans F. Design of experiments for quality improvement G. Auditing, ISO certification, and the Baldridge award

11%

EXAM SPECIFICATIONS—INDUSTRIAL

13

AFTERNOON SESSION IN MECHANICAL ENGINEERING (60 questions in 8 topic areas)

Topic Area


I.

Mechanical Design and Analysis A. Stress analysis (e.g., combined stresses, torsion, normal, shear) B. Failure theories (e.g., static, dynamic, buckling) C. Failure analysis (e.g., creep, fatigue, fracture, buckling) D. Deformation and stiffness E. Components (e.g., springs, pressure vessels, beams, piping, bearings, columns, power screws) F. Power transmission (e.g., belts, chains, clutches, gears, shafts, brakes, axles) G. Joining (e.g., threaded fasteners, rivets, welds, adhesives) H. Manufacturability (e.g., fits, tolerances, process capability) I. Quality and reliability J. Mechanical systems (e.g., hydraulic, pneumatic, electro-hybrid)

15%

II.

Kinematics, Dynamics, and Vibrations A. Kinematics of mechanisms B. Dynamics of mechanisms C. Rigid body dynamics D. Natural frequency and resonance E. Balancing of rotating and reciprocating equipment F. Forced vibrations (e.g., isolation, force transmission, support motion)

15%

III.

Materials and Processing A. Mechanical and thermal properties (e.g., stress/strain relationships, ductility, endurance, conductivity, thermal expansion) B. Manufacturing processes (e.g., forming, machining, bending, casting, joining, heat treating) C. Thermal processing (e.g., phase transformations, equilibria) D. Materials selection (e.g., metals, composites, ceramics, plastics, bio-materials) E. Surface conditions (e.g., corrosion, degradation, coatings, finishes) F. Testing (e.g., tensile, compression, hardness)

10%

IV.

Measurements, Instrumentation, and Controls A. Mathematical fundamentals (e.g., Laplace transforms, differential equations) B. System descriptions (e.g., block diagrams, ladder logic, transfer functions) C. Sensors and signal conditioning (e.g., strain, pressure, flow, force, velocity, displacement, temperature) D. Data collection and processing (e.g., sampling theory, uncertainty, digital/analog, data transmission rates) E. Dynamic responses (e.g., overshoot/time constant, poles and zeros, stability)

10%

V.

Thermodynamics and Energy Conversion Processes A. Ideal and real gases B. Reversibility/irreversibility C. Thermodynamic equilibrium D. Psychrometrics E. Performance of components F. Cycles and processes (e.g., Otto, Diesel, Brayton, Rankine) G. Combustion and combustion products H. Energy storage I. Cogeneration and regeneration/reheat

15%

14

EXAM SPECIFICATIONS—MECHANICAL

VI.

Fluid Mechanics and Fluid Machinery A. Fluid statics B. Incompressible flow C. Fluid transport systems (e.g., pipes, ducts, series/parallel operations) D. Fluid machines: incompressible (e.g., turbines, pumps, hydraulic motors) E. Compressible flow F. Fluid machines: compressible (e.g., turbines, compressors, fans) G. Operating characteristics (e.g., fan laws, performance curves, efficiencies, work/power equations) H. Lift/drag I. Impulse/momentum

15%

VII.

Heat Transfer A. Conduction B. Convection C. Radiation D. Composite walls and insulation E. Transient and periodic processes F. Heat exchangers G. Boiling and condensation heat transfer

10%

VIII.

Refrigeration and HVAC A. Cycles B. Heating and cooling loads (e.g., degree day data, sensible heat, latent heat) C. Psychrometric charts D. Coefficient of performance E. Components (e.g., compressors, condensers, evaporators, expansion valve)

10%

EXAM SPECIFICATIONS—MECHANICAL

15

AFTERNOON SESSION IN OTHER/GENERAL ENGINEERING (60 questions in 9 topic areas)

Topic Area


I.

Advanced Engineering Mathematics A. Differential equations B. Partial differential calculus C. Numerical solutions (e.g., differential equations, algebraic equations) D. Linear algebra E. Vector analysis

II.

Engineering Probability and Statistics A. Sample distributions and sizes B. Design of experiments C. Hypothesis testing D. Goodness of fit (coefficient of correlation, chi square) E. Estimation (e.g., point, confidence intervals) for two means

9%

III.

Biology A. Cellular biology (e.g., structure, growth, cell organization) B. Toxicology (e.g., human, environmental) C. Industrial hygiene [e.g., personnel protection equipment (PPE), carcinogens] D. Bioprocessing (e.g., fermentation, waste treatment, digestion)

5%

IV.

Engineering Economics A. Cost estimating B. Project selection C. Lease/buy/make D. Replacement analysis (e.g., optimal economic life)

10%

V.

Application of Engineering Mechanics A. Stability analysis of beams, trusses, and frames B. Deflection analysis C. Failure theory (e.g., static and dynamic) D. Failure analysis (e.g., creep, fatigue, fracture, buckling)

13%

VI.

Engineering of Materials A. Material properties of: 1. metals 2. plastics 3. composites 4. concrete

11%

VII.

Fluids A. Basic hydraulics (e.g., Manning equation, Bernoulli theorem, open-channel flow, pipe flow) B. Laminar and turbulent flow C. Friction losses (e.g., pipes, valves, fittings) D. Flow measurement E. Dimensionless numbers (e.g., Reynolds number) F. Fluid transport systems (e.g., pipes, ducts, series/parallel operations) G. Pumps, turbines, and compressors H. Lift/drag

15%

16

EXAM SPECIFICATIONS—GENERAL

10%

VIII.

Electricity and Magnetism A. Equivalent circuits (Norton, Thevenin) B. AC circuits (frequency domain) C. Network analysis (Kirchhoff laws) D. RLC circuits E. Sensors and instrumentation F. Electrical machines

12%

IX.

Thermodynamics and Heat Transfer A. Thermodynamic properties (e.g., entropy, enthalpy, heat capacity) B. Thermodynamic processes (e.g., isothermal, adiabatic, reversible, irreversible) C. Equations of state (ideal and real gases) D. Conduction, convection, and radiation heat transfer E. Mass and energy balances F. Property and phase diagrams (e.g., T-s, h-P) G. Tables of thermodynamic properties H. Cyclic processes and efficiency (e.g., refrigeration, power) I. Phase equilibrium and phase change J. Thermodynamic equilibrium K. Combustion and combustion products (e.g., CO, CO2, NOX, ash, particulates) L. Psychrometrics (e.g., humidity)

15%

EXAM SPECIFICATIONS—GENERAL

17

Do not write in this book or remove any pages. Do all scratch work in your exam booklet.

18

UNITS The FE exam and this handbook use both the metric system of units and the U.S. Customary System (USCS). In the USCS system of units, both force and mass are called pounds. Therefore, one must distinguish the pound-force (lbf) from the pound-mass (lbm). The pound-force is that force which accelerates one pound-mass at 32.174 ft/sec2. Thus, 1 lbf = 32.174 lbm-ft/sec2. The expression 32.174 lbm-ft/(lbf-sec2) is designated as gc and is used to resolve expressions involving both mass and force expressed as pounds. For instance, in writing Newton’s second law, the equation would be written as F = ma/gc, where F is in lbf, m in lbm, and a is in ft/sec2. Similar expressions exist for other quantities. Kinetic Energy, KE = mv2/2gc, with KE in (ft-lbf); Potential Energy, PE = mgh/gc, with PE in (ft-lbf); Fluid Pressure, p = ρgh/gc, with p in (lbf/ft2); Specific Weight, SW = ρg/gc, in (lbf/ft3); Shear Stress, τ = (μ/gc)(dv/dy), with shear stress in (lbf/ft2). In all these examples, gc should be regarded as a unit conversion factor. It is frequently not written explicitly in engineering equations. However, its use is required to produce a consistent set of units. Note that the conversion factor gc [lbm-ft/(lbf-sec2)] should not be confused with the local acceleration of gravity g, which has different units (m/s2 or ft/sec2) and may be either its standard value (9.807 m/s2 or 32.174 ft/sec2) or some other local value. If the problem is presented in USCS units, it may be necessary to use the constant gc in the equation to have a consistent set of units.

Multiple

METRIC PREFIXES Prefix

Symbol

10–18 10–15 10–12 10–9 10–6 10–3 10–2 10–1 101 102 103 106 109 1012 1015 1018

atto femto pico nano micro milli centi deci deka hecto kilo mega giga tera peta exa

a f p n μ m c d da h k M G T P E

COMMONLY USED EQUIVALENTS 1 gallon of water weighs

8.34 lbf

1 cubic foot of water weighs

62.4 lbf

1 cubic inch of mercury weighs

0.491 lbf

The mass of 1 cubic meter of water is

1,000 kilograms

TEMPERATURE CONVERSIONS ºF = 1.8 (ºC) + 32 ºC = (ºF – 32)/1.8 ºR = ºF + 459.69 K = ºC + 273.15

FUNDAMENTAL CONSTANTS Quantity electron charge Faraday constant gas constant gas constant gas constant gravitation - newtonian constant gravitation - newtonian constant gravity acceleration (standard) gravity acceleration (standard) molar volume (ideal gas), T = 273.15K, p = 101.3 kPa speed of light in vacuum Stephan-Boltzmann constant

metric metric USCS

metric USCS

Symbol e F R R R R G G g g Vm c σ

Value 1.6022 × 10−19 96,485 8,314 8.314 1,545 0.08206 6.673 × 10–11 6.673 × 10–11 9.807 32.174 22,414 299,792,000 5.67 × 10–8

Units C (coulombs) coulombs/(mol) J/(kmol•K) kPa•m3/(kmol•K) ft-lbf/(lb mole-ºR) L-atm/(mole-K) m3/(kg•s2) N•m2/kg2 m/s2 ft/sec2 L/kmol m/s W/(m2•K4) UNITS

19

CONVERSION FACTORS Multiply

By

To Obtain

Multiply 2

acre ampere-hr (A-hr) ångström (Å) atmosphere (atm) atm, std atm, std atm, std atm, std

43,560 3,600 1 × 10–10 76.0 29.92 14.70 33.90 1.013 × 105

square feet (ft ) coulomb (C) meter (m) cm, mercury (Hg) in, mercury (Hg) lbf/in2 abs (psia) ft, water pascal (Pa)

bar barrels–oil Btu Btu Btu Btu/hr Btu/hr Btu/hr

1 × 105 42 1,055 2.928 × 10–4 778 3.930 × 10–4 0.293 0.216

Pa gallons–oil joule (J) kilowatt-hr (kWh) ft-lbf horsepower (hp) watt (W) ft-lbf/sec

calorie (g-cal) cal cal cal/sec centimeter (cm) cm centipoise (cP) centipoise (cP) centistokes (cSt) cubic feet/second (cfs) cubic foot (ft3) cubic meters (m3) electronvolt (eV)

3.968 × 10–3 1.560 × 10–6 4.186 4.184 3.281 × 10–2 0.394 0.001 1 1 × 10–6 0.646317 7.481 1,000 1.602 × 10–19

Btu hp-hr joule (J) watt (W) foot (ft) inch (in) pascal•sec (Pa•s) g/(m•s) m2/sec (m2/s) million gallons/day (mgd) gallon Liters joule (J)

foot (ft) ft ft-pound (ft-lbf) ft-lbf ft-lbf ft-lbf

30.48 0.3048 1.285 × 10–3 3.766 × 10–7 0.324 1.356

cm meter (m) Btu kilowatt-hr (kWh) calorie (g-cal) joule (J)

ft-lbf/sec

1.818 × 10–3

horsepower (hp)

gallon (US Liq) gallon (US Liq) gallons of water gamma (γ, Γ) gauss gram (g)

3.785 0.134 8.3453 1 × 10–9 1 × 10–4 2.205 × 10–3

liter (L) ft3 pounds of water tesla (T) T pound (lbm)

hectare hectare horsepower (hp) hp hp hp hp-hr hp-hr hp-hr hp-hr

1 × 104 2.47104 42.4 745.7 33,000 550 2,545 1.98 × 106 2.68 × 106 0.746

square meters (m2) acres Btu/min watt (W) (ft-lbf)/min (ft-lbf)/sec Btu ft-lbf joule (J) kWh

inch (in) in of Hg in of Hg in of H2O in of H2O

2.540 0.0334 13.60 0.0361 0.002458

centimeter (cm) atm in of H2O lbf/in2 (psi) atm

20

CONVERSION FACTORS

By

To Obtain –4

joule (J) J J J/s

9.478 × 10 0.7376 1 1

kilogram (kg) kgf kilometer (km) km/hr kilopascal (kPa) kilowatt (kW) kW kW kW-hour (kWh) kWh kWh kip (K) K

2.205 9.8066 3,281 0.621 0.145 1.341 3,413 737.6 3,413 1.341 3.6 × 106 1,000 4,448

pound (lbm) newton (N) feet (ft) mph lbf/in2 (psi) horsepower (hp) Btu/hr (ft-lbf )/sec Btu hp-hr joule (J) lbf newton (N)

liter (L) L L L/second (L/s) L/s

61.02 0.264 10–3 2.119 15.85

in3 gal (US Liq) m3 ft3/min (cfm) gal (US)/min (gpm)

meter (m) m metric ton m/second (m/s) mile (statute) mile (statute) mile/hour (mph) mph mm of Hg mm of H2O

3.281 1.094 1,000 196.8 5,280 1.609 88.0 1.609 1.316 × 10–3 9.678 × 10–5

feet (ft) yard kilogram (kg) feet/min (ft/min) feet (ft) kilometer (km) ft/min (fpm) km/h atm atm

newton (N) newton (N) N•m N•m

0.225 1 0.7376 1

lbf kg•m/s2 ft-lbf joule (J)

pascal (Pa) Pa Pa•sec (Pa•s) pound (lbm, avdp) lbf lbf-ft lbf/in2 (psi) psi psi psi

9.869 × 10–6 1 10 0.454 4.448 1.356 0.068 2.307 2.036 6,895

atmosphere (atm) newton/m2 (N/m2) poise (P) kilogram (kg) N N •m atm ft of H2O in. of Hg Pa

radian

180/π

degree

stokes

1 × 10–4

m2/s

therm ton

1 × 105 2,000

Btu pounds (lb)

watt (W) W W weber/m2 (Wb/m2)

3.413 1.341 × 10–3 1 10,000

Btu/hr horsepower (hp) joule/s (J/s) gauss

Btu ft-lbf newton•m (N•m) watt (W)

MATHEMATICS STRAIGHT LINE The general form of the equation is Ax + By + C = 0 The standard form of the equation is y = mx + b, which is also known as the slope-intercept form. The point-slope form is y – y1 = m(x – x1) Given two points: slope, m = (y2 – y1)/(x2 – x1) The angle between lines with slopes m1 and m2 is α = arctan [(m2 – m1)/(1 + m2·m1)] Two lines are perpendicular if m1 = –1/m2 The distance between two points is d = _ y2 - y1i + ^ x2 - x1h 2

2

QUADRATIC EQUATION ax2 + bx + c = 0 b ! b 2 - 4ac x = Roots = 2a CONIC SECTIONS

Case 2. Ellipse e < 1: ♦

(x - h) 2 (y - k) 2 + = 1; Center at (h, k) a2 b2 is the standard form of the equation. When h = k = 0, Eccentricity:

e = 1 - _b 2/a 2i = c/a

b = a 1 - e 2; Focus: (± ae, 0); Directrix: x = ± a/e Case 3. Hyperbola e > 1: ♦

^ x - hh

a2

2

-

_ y - ki

b2

2

= 1; Center at (h, k)

e = eccentricity = cos θ/(cos φ) [Note: X ′ and Y ′, in the following cases, are translated axes.]

is the standard form of the equation.When h = k = 0,

Case 1. Parabola e = 1: ♦

b = a e 2 - 1;

Eccentricity: e = 1 + _b 2/a 2i = c/a Focus: (± ae, 0); Directrix: x = ± a/e

♦ Brink, R.W., A First Year of College Mathematics, D. Appleton-Century Co., Inc., 1937.

(y – k)2 = 2p(x – h); Center at (h, k) is the standard form of the equation. When h = k = 0, Focus: (p/2, 0); Directrix: x = –p/2

MATHEMATICS

21

Case 4. Circle e = 0: (x – h)2 + (y – k)2 = r2; Center at (h, k) is the standard form of the equation with radius

♦

2 2 r = ^ x - hh + _ y - k i

If a2 + b2 – c is positive, a circle, center (–a, –b). If a2 + b2 – c equals zero, a point at (–a, –b). If a2 + b2 – c is negative, locus is imaginary.

QUADRIC SURFACE (SPHERE) The standard form of the equation is (x – h)2 + (y – k)2 + (z – m)2 = r2 with center at (h, k, m). In a three-dimensional space, the distance between two points is 2 2 2 d = ^ x2 - x1h + _ y2 - y1i + ^ z2 - z1h

Length of the tangent from a point. Using the general form of the equation of a circle, the length of the tangent is found from t2 = (x′ – h)2 + (y′ – k)2 – r2 by substituting the coordinates of a point P(x′,y′) and the coordinates of the center of the circle into the equation and computing. ♦

LOGARITHMS The logarithm of x to the Base b is defined by logb (x) = c, where bc = x Special definitions for b = e or b = 10 are: ln x, Base = e log x, Base = 10 To change from one Base to another: logb x = (loga x)/(loga b) e.g., ln x = (log10 x)/(log10 e) = 2.302585 (log10 x) Identities logb bn = n log xc = c log x; xc = antilog (c log x) log xy = log x + log y logb b = 1; log 1 = 0 log x/y = log x – log y

Conic Section Equation The general form of the conic section equation is Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 where not both A and C are zero. If B2 – 4AC < 0, an ellipse is defined. If B2 – 4AC > 0, a hyperbola is defined. If B2 – 4AC = 0, the conic is a parabola. If A = C and B = 0, a circle is defined. If A = B = C = 0, a straight line is defined. x2 + y2 + 2ax + 2by + c = 0 is the normal form of the conic section equation, if that conic section has a principal axis parallel to a coordinate axis. h = –a; k = –b r = a2 + b2 - c

22

MATHEMATICS

TRIGONOMETRY Trigonometric functions are defined using a right triangle. sin θ = y/r, cos θ = x/r tan θ = y/x, cot θ = x/y csc θ = r/y, sec θ = r/x θ

Law of Sines a = b = c sin A sin B sin C Law of Cosines a2 = b2 + c2 – 2bc cos A b2 = a2 + c2 – 2ac cos B c2 = a2 + b2 – 2ab cos C

♦Brink, R.W., A First Year of College Mathematics, D. Appleton-Century Co., Inc., Englewood Cliffs, NJ, 1937.

Identities csc θ = 1/sin θ sec θ = 1/cos θ tan θ = sin θ/cos θ cot θ = 1/tan θ sin2θ + cos2θ = 1 tan2θ + 1 = sec2θ cot2θ + 1 = csc2θ sin (α + β) = sin α cos β + cos α sin β cos (α + β) = cos α cos β – sin α sin β sin 2α = 2 sin α cos α cos 2α = cos2α – sin2α = 1 – 2 sin2α = 2 cos2α – 1 tan 2α = (2 tan α)/(1 – tan2α) cot 2α = (cot2α – 1)/(2 cot α) tan (α + β) = (tan α + tan β)/(1 – tan α tan β) cot (α + β) = (cot α cot β – 1)/(cot α + cot β) sin (α – β) = sin α cos β – cos α sin β cos (α – β) = cos α cos β + sin α sin β tan (α – β) = (tan α – tan β)/(1 + tan α tan β) cot (α – β) = (cot α cot β + 1)/(cot β – cot α) sin (α/2) = ! ^1 - cos ah /2 cos (α/2) = ! ^1 + cos ah /2 tan (α/2) = ! ^1 - cos ah / ^1 + cos ah

cot (α/2) = ! ^1 + cos ah / ^1 - cos ah sin α sin β = (1/2)[cos (α – β) – cos (α + β)] cos α cos β = (1/2)[cos (α – β) + cos (α + β)] sin α cos β = (1/2)[sin (α + β) + sin (α – β)] sin α + sin β = 2 sin (1/2)(α + β) cos (1/2)(α – β) sin α – sin β = 2 cos (1/2)(α + β) sin (1/2)(α – β) cos α + cos β = 2 cos (1/2)(α + β) cos (1/2)(α – β) cos α – cos β = – 2 sin (1/2)(α + β) sin (1/2)(α – β)

COMPLEX NUMBERS Definition i = - 1 (a + ib) + (c + id) = (a + c) + i (b + d) (a + ib) – (c + id) = (a – c) + i (b – d) (a + ib)(c + id) = (ac – bd) + i (ad + bc) a + ib = â + ibh ^c - id h = âc + bd h + i ^bc - ad h c + id ^c + id h ^c - id h c2 + d2

Polar Coordinates x = r cos θ; y = r sin θ; θ = arctan (y/x) r = x + iy = x 2 + y 2 x + iy = r (cos θ + i sin θ) = reiθ [r1(cos θ1 + i sin θ1)][r2(cos θ2 + i sin θ2)] = r1r2[cos (θ1 + θ2) + i sin (θ1 + θ2)] n (x + iy) = [r (cos θ + i sin θ)]n = rn(cos nθ + i sin nθ) r1 _cos i1 + i sin i1i r = r1 8cos _i1 - i2i + i sin _i1 - i2iB 2 r2 _cos i2 + i sin i2i Euler’s Identity eiθ = cos θ + i sin θ e−iθ = cos θ – i sin θ ii ii - ii - ii cos i = e + e , sin i = e - e 2 2i Roots If k is any positive integer, any complex number (other than zero) has k distinct roots. The k roots of r (cos θ + i sin θ) can be found by substituting successively n = 0, 1, 2, ..., (k – 1) in the formula w=

k

r 1. Properties of Series n

! c = nc;

i=1 n

c = constant n

i=1

! _ xi + yi - zi i = ! xi + ! yi - ! zi

i=1 n

! x = _n + n 2i /2

Taylor’s Series f ^ x h = f ^ ah +

2 f l ^ ah ^ f m ^ ah ^ x - ah + x - ah 1! 2!

+ ... +

f

^ nh

^ ah

n!

^ x - ah + ... n

is called Taylor’s series, and the function f (x) is said to be expanded about the point a in a Taylor’s series. If a = 0, the Taylor’s series equation becomes a Maclaurin’s series.

DIFFERENTIAL CALCULUS The Derivative For any function y = f (x), the derivative = Dx y = dy/dx = y′ y l = limit 8_Dy i / ^DxhB Dx " 0

= limit $7 f ^ x + Dxh - f ^ xhA / ^Dxh. Dx " 0

y l = the slope of the curve f (x) .

Test for a Maximum y = f (x) is a maximum for x = a, if f ′(a) = 0 and f ″(a) < 0. Test for a Minimum y = f (x) is a minimum for x = a, if f ′(a) = 0 and f ″(a) > 0.

! cxi = c ! xi

i=1 n

4. Two power series may be added, subtracted, or multiplied, and the resulting series in each case is convergent, at least, in the interval common to the two series. 5. Using the process of long division (as for polynomials), two power series may be divided one by the other within their common interval of convergence.

n

n

n

i=1

i=1

i=1

x=1

Power Series ! i3= 0 ai ^ x - ah

i

1. A power series, which is convergent in the interval –R < x < R, defines a function of x that is continuous for all values of x within the interval and is said to represent the function in that interval. 2. A power series may be differentiated term by term within its interval of convergence. The resulting series has the same interval of convergence as the original series (except possibly at the end points of the series). 3. A power series may be integrated term by term provided the limits of integration are within the interval of convergence of the series.

Test for a Point of Inflection y = f (x) has a point of inflection at x = a, if f ″(a) = 0, and if f ″(x) changes sign as x increases through x = a. The Partial Derivative In a function of two independent variables x and y, a derivative with respect to one of the variables may be found if the other variable is assumed to remain constant. If y is kept fixed, the function z = f (x, y) becomes a function of the single variable x, and its derivative (if it exists) can be found. This derivative is called the partial derivative of z with respect to x. The partial derivative with respect to x is denoted as follows: 2f _ x, y i 2z 2x = 2x

MATHEMATICS

25

L’Hospital’s Rule (L’Hôpital’s Rule) If the fractional function f(x)/g(x) assumes one of the indeterminate forms 0/0 or ∞/∞ (where α is finite or infinite), then limit f ^ xh /g ^ xh

The Curvature of Any Curve ♦

x " a

is equal to the first of the expressions f l ^ xh f m ^ xh f n ^ xh limit , limit , limit x " a g l ^ xh x " a g m ^ xh x " a g n ^ xh which is not indeterminate, provided such first indicated limit exists. The curvature K of a curve at P is the limit of its average curvature for the arc PQ as Q approaches P. This is also expressed as: the curvature of a curve at a given point is the rate-of-change of its inclination with respect to its arc length.

Curvature in Rectangular Coordinates ym K= 3 2 91 + _ y li2C When it may be easier to differentiate the function with respect to y rather than x, the notation x′ will be used for the derivative. x l = dx/dy - xm K= 3 2 81 + ^ x lh2B The Radius of Curvature The radius of curvature R at any point on a curve is defined as the absolute value of the reciprocal of the curvature K at that point. ^ K ! 0h R= 1 K 3 2

26

MATHEMATICS

limit ! f _ xi i Dxi = #a f ^ xh dx n

b

n " 3i=1

Also, Δxi →0 for all i.

K = limit Da = da ds Ds " 0 Ds

91 + _ y li2C R= ym

INTEGRAL CALCULUS The definite integral is defined as:

_ y m ! 0i

A table of derivatives and integrals is available in the Derivatives and Indefinite Integrals section. The integral equations can be used along with the following methods of integration: A. Integration by Parts (integral equation #6), B. Integration by Substitution, and C. Separation of Rational Fractions into Partial Fractions.

♦ Wade, Thomas L., Calculus, Ginn & Company/Simon & Schuster Publishers, 1953.

DERIVATIVES AND INDEFINITE INTEGRALS In these formulas, u, v, and w represent functions of x. Also, a, c, and n represent constants. All arguments of the trigonometric functions are in radians. A constant of integration should be added to the integrals. To avoid terminology difficulty, the following definitions are followed: arcsin u = sin–1 u, (sin u)–1 = 1/sin u. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

dc/dx = 0 dx/dx = 1 d(cu)/dx = c du/dx d(u + v – w)/dx = du/dx + dv/dx – dw/dx d(uv)/dx = u dv/dx + v du/dx d(uvw)/dx = uv dw/dx + uw dv/dx + vw du/dx d û/vh = v du/dx -2 u dv/dx dx v n n–1 d(u )/dx = nu du/dx d[f (u)]/dx = {d[f (u)]/du} du/dx du/dx = 1/(dx/du) d _logaui = _logaei u1 du dx dx

^ h 12. d 1n u = u1 du dx dx ui _ 13. d a = ^1n ah au du dx dx u u 14. d(e )/dx = e du/dx

15. 16. 17. 18. 19. 20. 21.

d(uv)/dx = vuv–1 du/dx + (ln u) uv dv/dx d(sin u)/dx = cos u du/dx d(cos u)/dx = –sin u du/dx d(tan u)/dx = sec2u du/dx d(cot u)/dx = –csc2u du/dx d(sec u)/dx = sec u tan u du/dx d(csc u)/dx = –csc u cot u du/dx

_ -1 i du 1 22. d sin u = 2 dx dx 1-u

_ -1 i du 1 23. d cos u =2 dx dx 1-u _ tan- 1ui d du 1 24. = dx 1 + u 2 dx _ -1 i 25. d cot u =- 1 2 du dx 1 + u dx -1 i _ du 1 26. d sec u = dx u u 2 - 1 dx

27.

_- r/2 # sin- 1u # r/2i _0 # cos- 1u # ri _- r/2 < tan- 1u < r/2i _0 < cot u < ri

dx

5. 6. 7. 8.

# d f (x) = f (x) # dx = x # a f(x) dx = a # f(x) dx # [u(x) ± v(x)] dx = # u(x) dx ± # v(x) dx m+1 # x mdx = x ^m ! - 1h m+1 # u(x) dv(x) = u(x) v(x) – # v (x) du(x) # dx = a1 1n ax + b ax + b # dx = 2 x x

x # a x dx = a 1na 10. # sin x dx = – cos x 11. # cos x dx = sin x

9.

12. # sin 2xdx = x - sin 2x 2 4 2 x sin 2x 13. # cos xdx = + 2 4 14. # x sin x dx = sin x – x cos x 15. # x cos x dx = cos x + x sin x 16. # sin x cos x dx = (sin2x)/2 17. # sin ax cos bx dx =- cos â - bh x - cos â + bh x _a 2 ! b 2i 2 â - bh 2 â + bh 18. # tan x dx = –ln⏐cos x⏐= ln ⏐sec x⏐ 19. # cot x dx = –ln ⏐csc x ⏐= ln ⏐sin x⏐ 20. # tan2x dx = tan x – x 21. # cot2x dx = –cot x – x 22. # eax dx = (1/a) eax 23. # xeax dx = (eax/a2)(ax – 1) 24. # ln x dx = x [ln (x) – 1] (x > 0)

â ! 0h

25. #

dx = 1 tan- 1 x a a a2 + x2

26. #

dx = 1 tan- 1 b x ac l, ac ax + c

27a. #

dx = ax 2 + bx + c

2 tan- 1 2ax + b 2 4ac - b 2 4ac - b

dx = ax + bx + c

2ax + b - b 2 - 4ac 1 1 n b 2 - 4ac 2ax + b + b 2 - 4ac

-1

_0 < sec- 1u < r/2i _- r # sec- 1u < - r/2i d _csc- 1ui du 1

=-

1. 2. 3. 4.

u u - 1 dx

2

_ 4ac - b 2 > 0i

2

_0 < csc u # r/2i _- r < csc u # - r/2i -1

-1

27b. #

27c. #

2

â > 0, c > 0h

dx 2 , =2ax + b ax 2 + bx + c

_b 2 - 4ac > 0i

_b 2 - 4ac = 0i

MATHEMATICS

27

MENSURATION OF AREAS AND VOLUMES

Circular Sector ♦

Nomenclature A = total surface area P = perimeter V = volume Parabola

A = zr 2/2 = sr/2 z = s/r Sphere ♦

Ellipse ♦

V = 4rr3/3 = rd3/6 A = 4rr 2 = rd 2 Parallelogram Papprox = 2r _a 2 + b 2i /2 R 2 2 S1 + _ 1 2i m 2 + _ 1 2 # 1 4i m 4 S 2 P = r â + bh S+ _ 1 2 # 1 4 # 3 6 i m6 + _ 1 2 # 1 4 # 3 6 # S SS+ _ 1 # 1 # 3 # 5 # 7 i2 m10 + f 2 4 6 8 10 T where

m = â - bh / â + bh

V W W 2 5 i m8 W, 8 W WW X

P = 2 â + bh

d1 = a 2 + b 2 - 2ab ^cos zh

d2 = a 2 + b 2 + 2ab ^cos zh d12 + d22 = 2 _a 2 + b 2i

Circular Segment ♦

A = ah = ab ^sin zh

A

If a = b, the parallelogram is a rhombus. ♦Gieck, K. & R. Gieck, Engineering Formulas, 6th ed., Gieck Publishing, 1967.

A = 8 r 2 ^z - sin zhB /2

z = s/r = 2 $arccos 7^r - d h /r A.

28

MATHEMATICS

MENSURATION OF AREAS AND VOLUMES (continued)

Right Circular Cone ♦

Regular Polygon (n equal sides) ♦

z = 2r/n

h ^ i = ; r n n- 2 E = r b1 - n2 l P = ns

s = 2r 8 tan ^z/2hB A = ^nsr h /2

V = _rr 2hi /3 A = side area + base area = rr ` r + r 2 + h 2 j Ax: Ab = x 2: h 2

Right Circular Cylinder ♦

Prismoid ♦

2 V = rr 2h = rd h 4 A = side area + end areas = 2rr ^h + r h

V = ^h/6h ^ A1 + A2 + 4Ah

Paraboloid of Revolution

2 V = rd h 8

♦Gieck, K. & R. Gieck, Engineering Formulas, 6th ed., Gieck Publishing, 1967.

MATHEMATICS

29

CENTROIDS AND MOMENTS OF INERTIA The location of the centroid of an area, bounded by the axes and the function y = f(x), can be found by integration. # xdA A # ydA yc = A A = # f ^ xh dx

xc =

The first moment of area with respect to the y-axis and the x-axis, respectively, are: My = ∫x dA = xc A Mx = ∫y dA = yc A The moment of inertia (second moment of area) with respect to the y-axis and the x-axis, respectively, are: Iy = ∫x2 dA Ix = ∫y2 dA The moment of inertia taken with respect to an axis passing through the area’s centroid is the centroidal moment of inertia. The parallel axis theorem for the moment of inertia with respect to another axis parallel with and located d units from the centroidal axis is expressed by Iparallel axis = Ic + Ad2 In a plane, J =∫r2dA = Ix + Iy Values for standard shapes are presented in tables in the STATICS and DYNAMICS sections.

DIFFERENTIAL EQUATIONS A common class of ordinary linear differential equations is dy ^ xh d n y ^ xh + f + b1 + b0 y ^ xh = f ^ xh n dx dx

where bn, … , bi, … , b1, b0 are constants. When the equation is a homogeneous differential equation, f(x) = 0, the solution is

yh ^ xh = C1e r1 x + C2e r 2 x + f + Cie r i x + f + Cne r n x

where rn is the nth distinct root of the characteristic polynomial P(x) with r2 x

If the root r1 = r2, then C2e

r1x

is replaced with C2xe

.

Higher orders of multiplicity imply higher powers of x. The complete solution for the differential equation is y(x) = yh(x) + yp(x), where yp(x) is any particular solution with f(x) present. If f(x) has ern x terms, then resonance is manifested. Furthermore, specific f(x) forms result in specific yp(x) forms, some of which are:

MATHEMATICS

Aeαx

Beαx, α ≠rn

A1 sin ωx + A2 cos ωx

B1 sin ωx + B2 cos ωx

B

First-Order Linear Homogeneous Differential Equations with Constant Coefficients y′+ ay = 0, where a is a real constant: Solution, y = Ce–at where C = a constant that satisfies the initial conditions. First-Order Linear Nonhomogeneous Differential Equations A t0 y ^0h = KA

τ is the time constant K is the gain The solution is

y ^t h = KA + ] KB - KAg c1 - exp b -xt lm or

t KB - KA x = 1n < KB - y F

Second-Order Linear Homogeneous Differential Equations with Constant Coefficients An equation of the form y″+ ay′+ by = 0 can be solved by the method of undetermined coefficients where a solution of the form y = Cerx is sought. Substitution of this solution gives (r2 + ar + b) Cerx = 0 and since Cerx cannot be zero, the characteristic equation must vanish or r2 + ar + b = 0 The roots of the characteristic equation are r1, 2 =-

a ! a 2 - 4b 2

and can be real and distinct for a2 > 4b, real and equal for a2 = 4b, and complex for a2 < 4b.

P(r) = bnrn + bn–1rn–1 + … + b1r + b0

30

yp(x)

If the independent variable is time t, then transient dynamic solutions are implied.

dA = f ^ xh dx = g _ y i dy

bn

f(x) A

If a2 > 4b, the solution is of the form (overdamped) y = C1er1x + C2er2 x If a2 = 4b, the solution is of the form (critically damped) y = (C1+ C2x)er1x If a2 < 4b, the solution is of the form (underdamped) y = eαx (C1 cos βx + C2 sin βx), where α= – a/2 b=

4b - a 2 2

FOURIER TRANSFORM The Fourier transform pair, one form of which is 3 F ^~h = #- 3 f ^t h e- j~t dt

NUMERICAL METHODS Newton’s Method for Root Extraction Given a function f(x) which has a simple root of f(x) = 0 at x = a an important computational task would be to find that root. If f(x) has a continuous first derivative then the (j +1)st estimate of the root is

f ^t h = 71/ ^2rhA #- 3 F ^~h e j~t d~ 3

can be used to characterize a broad class of signal models in terms of their frequency or spectral content. Some useful transform pairs are: f(t) F(ω) d ^t h 1

u(t) u bt + x l - u bt - x l = rrect xt 2 2 e

j~o t

rd ^~h + 1/j~ ^

h

x sin ~x/2 ~x/2

2rd _~ - ~oi

Some mathematical liberties are required to obtain the second and fourth form. Other Fourier transforms are derivable from the Laplace transform by replacing s with jω provided f ^t h = 0, t < 0 3

#0

f ^t h dt < 3

Also refer to Fourier Series and Laplace Transforms in the ELECTRICAL AND COMPUTER ENGINEERING section of this handbook.

DIFFERENCE EQUATIONS Difference equations are used to model discrete systems. Systems which can be described by difference equations include computer program variables iteratively evaluated in a loop, sequential circuits, cash flows, recursive processes, systems with time-delay components, etc. Any system whose input v(t) and output y(t) are defined only at the equally spaced intervals t = kT can be described by a difference equation. First-Order Linear Difference Equation The difference equation Pk = Pk–1(1 + i) – A represents the balance P of a loan after the kth payment A. If Pk is defined as y(k), the model becomes y(k) – (1 + i) y(k – 1) = – A

f ^ xh df ^ xh dx x a j = The initial estimate of the root a0 must be near enough to the actual root to cause the algorithm to converge to the root. aj+1 = aj -

Newton’s Method of Minimization Given a scalar value function h(x) = h(x1, x2, …, xn) find a vector x*∈Rn such that h(x*) ≤ h(x) for all x Newton’s algorithm is J N- 1 K 2 O O 2h xk + 1 = xk - K 2 h2 K 2x O 2x x x = k L P R V 2 h S W S 2x1 W S 2h W S 2x W 2h = S 2 W Sg W 2x S W Sg W S 2h W S 2x W n T X and R 2 2 2h S2 h g 2 2x1 2x2 S 2x1 S S 2 2h 2 2h g S 2x 2x 2x22 2 2h = S 1 2 Sg g g 2x 2 S g g Sg S 2 2 2h S 2h g 2x2 2xn S 2x1 2xn T

, where x = xk

V 2 2h W 2x1 2xn W W 2 W h 2 g 2x2 2xn W W g g W W g g W W 2 2h W g 2xn2 W X g

Second-Order Linear Difference Equation The Fibonacci number sequence can be generated by y(k) = y(k – 1) + y(k – 2) where y(–1) = 1 and y(–2) = 1. An alternate form for this model is f (k + 2) = f (k + 1) + f (k) with f (0) = 1 and f (1) = 1.

MATHEMATICS

31

Numerical Integration Three of the more common numerical integration algorithms used to evaluate the integral b #a f ^ xh dx are: Euler’s or Forward Rectangular Rule

#a f ^ xh dx . Dx ! f â + kDxh b

n-1

k=0

Trapezoidal Rule for n = 1 #a f ^ xh dx . Dx < b

for n > 1

f âh + f ^bh F 2

n-1 b #a f ^ xh dx . Dx < f âh + 2 ! f â + kDxh + f ^bhF 2 k=1

Simpson’s Rule/Parabolic Rule (n must be an even integer) for n = 2 b #a f ^ xh dx . b b - a l ; f âh + 4f b a + b l + f ^bhE 6 2 for n ≥ 4

R V n-2 S f ^ ah + 2 ! f â + kDxh W S W k = 2, 4, 6, f b x D S W ^ h #a f x dx . W 3 S n-1 S 4 ^ h ^ h f a + kDx + f b WW S+ k = 1! , 3, 5, f T X

with Δx = (b – a)/n n = number of intervals between data points Numerical Solution of Ordinary Differential Equations

Euler’s Approximation Given a differential equation dx/dt = f (x, t) with x(0) = xo At some general time kΔt x[(k + 1)Δt] ≅ x(kΔt) + Δtf [x(kΔt), kΔt] which can be used with starting condition xo to solve recursively for x(Δt), x(2Δt), …, x(nΔt). The method can be extended to nth order differential equations by recasting them as n first-order equations. In particular, when dx/dt = f (x) x[(k + 1)Δt] ≅ x(kΔt) + Δtf [x(kΔt)] which can be expressed as the recursive equation xk + 1 = xk + Δt (dxk/dt) Refer to the ELECTRICAL AND COMPUTER ENGINEERING section for additional information on Laplace transforms and algebra of complex numbers.

32

MATHEMATICS

MECHANICS OF MATERIALS Uniaxial Loading and Deformation

UNIAXIAL STRESS-STRAIN

σ

=

σ = P/A, where stress on the cross section,

P

=

loading, and

A

=

cross-sectional area.

δ

=

ε = δ/L, where elastic longitudinal deformation and

L

=

length of member.

Stress-Strain Curve for Mild Steel

STRESS, MPa

STRESS, PSI

♦

E=v f=

P A d L

d = PL AE The slope of the linear portion of the curve equals the modulus of elasticity.

True stress is load divided by actual cross-sectional area whereas engineering stress is load divided by the initial area.

DEFINITIONS

THERMAL DEFORMATIONS δt = αL(T – To), where

Engineering Strain ε = ΔL/Lo, where

δt

=

deformation caused by a change in temperature,

α

=

temperature coefficient of expansion,

L

=

length of member,

T

=

final temperature, and

ε

=

engineering strain (units per unit),

ΔL =

change in length (units) of member,

Lo =

original length (units) of member.

To =

initial temperature.

Percent Elongation

% Elongation = c DL m # 100 Lo

Percent Reduction in Area (RA) The % reduction in area from initial area, Ai, to final area, Af , is: A - Af o # 100 %RA = e i Ai Shear Stress-Strain γ = τ/G, where

γ

=

shear strain,

τ

=

shear stress, and

G =

shear modulus (constant in linear torsion-rotation relationship). G=

E v

= = =

E , where 2 ^1 + vh

modulus of elasticity Poisson’s ratio, and – (lateral strain)/(longitudinal strain).

CYLINDRICAL PRESSURE VESSEL Cylindrical Pressure Vessel For internal pressure only, the stresses at the inside wall are: vt = Pi

ro2 + ri2 and vr =-Pi ro2 - ri2

For external pressure only, the stresses at the outside wall are: vt =-Po

ro2 + ri2 and vr =-Po , where ro2 - ri2

σt =

tangential (hoop) stress,

σr =

radial stress,

Pi =

internal pressure,

Po =

external pressure,

=

inside radius, and

ri

ro =

outside radius.

For vessels with end caps, the axial stress is: va = Pi

ri2 ro2 - ri2

σt, σr, and σa are principal stresses. ♦ Flinn, Richard A. & Paul K. Trojan, Engineering Materials & Their Applications, 4th ed., Houghton Mifflin Co., Boston, 1990.

MECHANICS OF MATERIALS

33

When the thickness of the cylinder wall is about one-tenth or less of inside radius, the cylinder can be considered as thinwalled. In which case, the internal pressure is resisted by the hoop stress and the axial stress. Pr vt = ti

Pr and va = i 2t

The circle drawn with the center on the normal stress (horizontal) axis with center, C, and radius, R, where vx + v y vx - v y n + x 2xy C= , R= d 2 2 2

The two nonzero principal stresses are then: ♦

where t = wall thickness.

cw

STRESS AND STRAIN Principal Stresses For the special case of a two-dimensional stress state, the equations for principal stress reduce to

σa = C + R σb = C – R y,

xy R

vx + v y vx - v y n + x 2xy ! d 2 2 2

va, vb =

R

in

a

b C

2

vc = 0 x, xy

The two nonzero values calculated from this equation are temporarily labeled σa and σb and the third value σc is always zero in this case. Depending on their values, the three roots are then labeled according to the convention: algebraically largest = σ1, algebraically smallest = σ3, other = σ2. A typical 2D stress element is shown below with all indicated components shown in their positive sense. ♦ To

ccw

The maximum inplane shear stress is τin = R. However, the maximum shear stress considering three dimensions is always xmax =

v1 - v3 . 2

Hooke's Law Three-dimensional case: εx = (1/E)[σx – v(σy+ σz)]

γxy = τxy /G

εy = (1/E)[σy – v(σz+ σx)]

γyz = τyz /G

εz = (1/E)[σz – v(σx+ σy)]

γzx = τzx /G

Plane stress case (σz = 0): εx = (1/E)(σx – vσy) εy = (1/E)(σy – vσx) Mohr’s Circle – Stress, 2D To construct a Mohr’s circle, the following sign conventions are used. 1. Tensile normal stress components are plotted on the horizontal axis and are considered positive. Compressive normal stress components are negative. 2. For constructing Mohr’s circle only, shearing stresses are plotted above the normal stress axis when the pair of shearing stresses, acting on opposite and parallel faces of an element, forms a clockwise couple. Shearing stresses are plotted below the normal axis when the shear stresses form a counterclockwise couple.

εz = – (1/E)(vσx + vσy) Uniaxial case (σy = σz = 0): εx, εy, εz = normal strain,

R vx 0 S1 v E S v v 1 0 * y 4 = 1 - v2 S 1 xxy S0 0 2 T

V W fx W* f y 4 vW c W xy X

σx = Eεx or σ = Eε, where

σx, σy, σz = normal stress, γxy, γyz, γzx = shear strain, τxy, τyz, τzx = shear stress, E = modulus of elasticity, G = shear modulus, and v = Poisson’s ratio.

♦ Crandall, S.H. and N.C. Dahl, An Introduction to Mechanics of Solids, McGraw-Hill, New York, 1959.

34


STATIC LOADING FAILURE THEORIES See MATERIALS SCIENCE/STRUCTURE OF MATTER for Stress Concentration in Brittle Materials. Brittle Materials Maximum-Normal-Stress Theory The maximum-normal-stress theory states that failure occurs when one of the three principal stresses equals the strength of the material. If σ1 ≥ σ2 ≥ σ3, then the theory predicts that failure occurs whenever σ1 ≥ Sut or σ3 ≤ – Suc where Sut and Suc are the tensile and compressive strengths, respectively. Coulomb-Mohr Theory The Coulomb-Mohr theory is based upon the results of tensile and compression tests. On the σ, τ coordinate system, one circle is plotted for Sut and one for Suc. As shown in the figure, lines are then drawn tangent to these circles. The CoulombMohr theory then states that fracture will occur for any stress situation that produces a circle that is either tangent to or crosses the envelope defined by the lines tangent to the Sut and Suc circles. τ

-S uc

σ3

Distortion-Energy Theory The distortion-energy theory states that yielding begins whenever the distortion energy in a unit volume equals the distortion energy in the same volume when uniaxially stressed to the yield strength. The theory predicts that yielding will occur whenever

= ^v1 - v2h + _v2 - v3i + _v1 - v3i G 2 2

2

2

1 2

$ Sy

The term on the left side of the inequality is known as the effective or Von Mises stress. For a biaxial stress state the effective stress becomes vl = `v 2A - vA vB + v 2Bj

1 2

or

vl = `v 2x - vx v y + v 2y + 3x 2xy j

1 2

where σA and σB are the two nonzero principal stresses and σx, σy, and τxy are the stresses in orthogonal directions.

VARIABLE LOADING FAILURE THEORIES Modified Goodman Theory: The modified Goodman criterion states that a fatigue failure will occur whenever σ1

S ut

va vm $ 1 or Se + Sut

σ

vmax $ 1, vm $ 0, Sy

where Se = fatigue strength,

If σ1 ≥ σ2 ≥ σ3 and σ3 < 0, then the theory predicts that yielding will occur whenever v3 v1 $ 1 Sut - Suc Ductile Materials Maximum-Shear-Stress Theory The maximum-shear-stress theory states that yielding begins when the maximum shear stress equals the maximum shear stress in a tension-test specimen of the same material when that specimen begins to yield. If σ1 ≥ σ2 ≥ σ3, then the theory predicts that yielding will occur whenever τmax ≥ Sy /2 where Sy is the yield strength. v - v3 . xmax = 1 2

Sut

= ultimate strength,

Sy

= yield strength,

σa

= alternating stress, and

σm = mean stress. σmax = σm + σa Soderberg Theory: The Soderberg theory states that a fatigue failure will occur whenever va vm $ 1 vm $ 0 Se + Sy Endurance Limit for Steels: When test data is unavailable, the endurance limit for steels may be estimated as S le = *

0.5 Sut, Sut # 1, 400 MPa 4 700 MPa, Sut > 1, 400 MPa


35

Endurance Limit Modifying Factors: Endurance limit modifying factors are used to account for the differences between the endurance limit as determined from a rotating beam test, S le , and that which would result in the real part, Se. Se = ka kb kc kd ke S le

TORSIONAL STRAIN czz = limit r ^Dz/Dz h = r ^dz/dz h Dz " 0

The shear strain varies in direct proportion to the radius, from zero strain at the center to the greatest strain at the outside of the shaft. dφ/dz is the twist per unit length or the rate of twist. xzz = Gczz = Gr ^dz/dz h

where Surface Factor, ka = aSutb Surface Finish Ground Machined or CD Hot rolled As forged

Factor a kpsi 1.34

MPa 1.58

2.70

4.51

14.4 39.9

Size Factor, kb: For bending and torsion: d ≤ 8 mm;

57.7 272.0

z = #o T dz = TL , where GJ GJ L

–0.085 –0.265 –0.718 –0.995

kb = 1 kb = 1.189d -eff0.097

d > 250 mm;

0.6 ≤ kb ≤ 0.75

Load Factor, kc: kc = 0.923

A

Exponent b

8 mm ≤ d ≤ 250 mm; For axial loading:

T = G ^dz/dz h # r 2dA = GJ ^dz/dz h

kb = 1 axial loading, Sut ≤ 1,520 MPa

φ = total angle (radians) of twist, T = torque, and L = length of shaft. T/φ gives the twisting moment per radian of twist. This is called the torsional stiffness and is often denoted by the symbol k or c. For Hollow, Thin-Walled Shafts x = T , where 2Amt t Am

= thickness of shaft wall and = the total mean area enclosed by the shaft measured to the midpoint of the wall.

kc = 1

axial loading, Sut > 1,520 MPa

BEAMS

kc = 1

bending

kc = 0.577

torsion

Shearing Force and Bending Moment Sign Conventions 1. The bending moment is positive if it produces bending of the beam concave upward (compression in top fibers and tension in bottom fibers).

Temperature Factor, kd: for T ≤ 450°C, kd = 1 Miscellaneous Effects Factor, ke: Used to account for strength reduction effects such as corrosion, plating, and residual stresses. In the absence of known effects, use ke = 1.

TORSION Torsion stress in circular solid or thick-walled (t > 0.1 r) shafts: x = Tr J

2. The shearing force is positive if the right portion of the beam tends to shear downward with respect to the left. ♦

POSITIVE BENDING

NEGATIVE BENDING

POSITIVE SHEAR

NEGATIVE SHEAR

where J = polar moment of inertia (see table at end of STATICS section). ♦ Timoshenko, S. and Gleason H. MacCullough, Elements of Strengths of Materials, K. Van Nostrand Co./Wadsworth Publishing Co., 1949.

36


The relationship between the load (q), shear (V), and moment (M) equations are: ^ h q ^ xh =- dV x dx ^ h V = dM x dx

Deflection of Beams Using 1/ρ = M/(EI), d 2y = M, differential equation of deflection curve dx 2 d3y EI 3 = dM ^ xh /dx = V dx d 4y EI 4 = dV ^ xh /dx =- q dx EI

V2 - V1 = #x 2 7- q ^ xhA dx x

1

M2 - M1 = #x 2 V ^ xh dx x

1

Stresses in Beams εx = – y/ρ, where ρ

= the radius of curvature of the deflected axis of the beam, and

y

= the distance from the neutral axis to the longitudinal fiber in question.

Using the stress-strain relationship σ = Eε, Axial Stress: σx = –Ey/ρ, where σx

= the normal stress of the fiber located y-distance from the neutral axis.

Determine the deflection curve equation by double integration (apply boundary conditions applicable to the deflection and/or slope). EI (dy/dx) = ∫M(x) dx EIy = ∫[ ∫M(x) dx] dx The constants of integration can be determined from the physical geometry of the beam.

COLUMNS For long columns with pinned ends: Euler’s Formula 2 Pcr = r EI , where ,2

1/ρ = M/(EI), where M

= the moment at the section and

I

= the moment of inertia of the cross section. σx = – My/I, where

y

= the distance from the neutral axis to the fiber location above or below the axis. Let y = c, where c = distance from the neutral axis to the outermost fiber of a symmetrical beam section. σx = ± Mc/I

Let S = I/c: then, σx = ± M/S, where S

=

Pcr =

critical axial loading,

,

unbraced column length.

=

substitute I = r2A: Pcr r 2E , where A = ^,/r h2 r

=

,/r =

radius of gyration and slenderness ratio for the column.

For further column design theory, see the CIVIL ENGINEERING and MECHANICAL ENGINEERING sections.

the elastic section modulus of the beam member.

Transverse shear flow:

q = VQ/I and

Transverse shear stress: τxy = VQ/(Ib), where q

=

shear flow,

τxy =

shear stress on the surface,

V

=

shear force at the section,

b

=

width or thickness of the cross-section, and

Q =

Al y l , where

A′ =

area above the layer (or plane) upon which the desired transverse shear stress acts and

yl =

distance from neutral axis to area centroid.


37

ELASTIC STRAIN ENERGY If the strain remains within the elastic limit, the work done during deflection (extension) of a member will be transformed into potential energy and can be recovered. If the final load is P and the corresponding elongation of a tension member is δ, then the total energy U stored is equal to the work W done during loading. U = W = Pδ/2

The strain energy per unit volume is u = U/AL = σ2/2E

(for tension)

Cast Iron

Wood (Fir)

Modulus of Rigidity, G

Aluminum

Modulus of Elasticity, E

Mpsi

29.0

10.0

14.5

1.6

GPa

200.0 69.0

100.0

11.0

Mpsi

11.5

3.8

6.0

0.6

GPa

80.0

26.0

41.4

4.1

0.30

0.33

0.21

0.33

10−6 °F

6.5

13.1

6.7

1.7

10−6 °C

11.7

23.6

12.1

3.0

Units

Material

Steel

MATERIAL PROPERTIES

Poisson's Ratio, v Coefficient of Thermal Expansion, α

38



39

a

L

P

b

y

L

P

R1

L

δ max φ max

R2 = w L/2

w (LOAD PER UNIT LENGTH)

L

x

2

2

x

M2

M x2 2 EI

[

(

w x2 2 x + 6 L2 − 4 Lx 24 EI

)

(

)]

[

(

)]

)

w x2 2 (L − Lx + x2) 24EI

(

wx L3 − 2 Lx 2 + x 3 24 EI

Pb − x 3 + L2 − b 2 x , for x ≤ a 6 LEI

δ(x) =

δ=

δ=

Pb L (x − a )3 − x 3 + L2 − b 2 x , for x > a δ= 6 LEI b

δ=

δ=

Pa 2 (3x − a ), for x > a 6 EI Px 2 (− x + 3a ), for x ≤ a δ= 6 EI δ=

Crandall, S.H. and N.C. Dahl, An Introduction to Mechanics of Solids, McGraw-Hill, New York, 1959.

2

R2

R2 = Pa/L

b

x φmax

δmax

x φmax

δmax

x φmax

δmax

R1 = R 2 = w L and M1 = M 2 = w L 2 12

M1

a

L

M

w (LOAD PER UNIT LENGTH)

R1 = w L/2

1

y

R1 = Pb/L

1

y

L

y w (LOAD PER UNIT LENGTH)

y

4

5 wL 384 EI

at x =

9 3 LEI

(

Pb L2 − b 2

M L2 2 EI

w L4 8EI

L2 − b 2 3

32

)

Pa 2 (3L − a ) 6 EI

4 L δ max = wL at x = 384 EI 2

δ max =

δ max =

δ max =

δ max =

δ max =

Beam Deflection Formulas – Special Cases (δ is positive downward)

wL3 24 EI l L at x = ± 2 12

φ max = 0.008

w L3 24 EI

Pab(2 L − b ) 6 LEI

Pab(2 L − a ) 6 LEI

ML EI

w L3 6 EI

Pa 2 2 EI

φ1 = φ 2 =

φ2 =

φ1 =

φ max =

φ max =

φ max =

ENGINEERING PROBABILITY AND STATISTICS DISPERSION, MEAN, MEDIAN, AND MODE VALUES If X1, X2, … , Xn represent the values of a random sample of n items or observations, the arithmetic mean of these items or observations, denoted X , is defined as X = _1 ni _ X1 + X2 + f + Xni = _1 ni ! Xi n

i=1

X " n for sufficiently large values of n.

nPr is an alternative notation for P(n,r)

The weighted arithmetic mean is !w X X w = ! i i , where wi Xi = the value of the ith observation, and wi = the weight applied to Xi. The variance of the population is the arithmetic mean of the squared deviations from the population mean. If μ is the arithmetic mean of a discrete population of size N, the population variance is defined by

2 2 2 v 2 = ^1/N h 9^ X1 - nh + ^ X2 - nh + f + _ XN - ni C

= ^1/N h ! _ Xi - ni N

PERMUTATIONS AND COMBINATIONS A permutation is a particular sequence of a given set of objects. A combination is the set itself without reference to order. 1. The number of different permutations of n distinct objects taken r at a time is n! P ^n, r h = ^n - r h !

2

2. The number of different combinations of n distinct objects taken r at a time is P ^ n, r h n! C ^n, r h = r! = 7 r! ^n - r h !A

n nCr and e o are alternative notations for C(n,r) r 3. The number of different permutations of n objects taken n at a time, given that ni are of type i, where i = 1, 2, …, k and ∑ni = n, is n! P _ n; n1, n2, f, nk i = n1!n2!fnk!

SETS

i=1

The standard deviation of the population is v = ^1/N h ! _ Xi - ni

DeMorgan’s Law

2

A,B=A+B A+B=A,B

The sample variance is

s 2 = 71/ ^n - 1hA ! ` Xi - X j n

2

Associative Law

A , ^ B , C h = ] A , Bg , C A + ^ B + C h = ] A + Bg + C

i=1

The sample standard deviation is

s = 71/ ^n - 1hA ! ` Xi - X j n

2

i=1

Distributive Law

A , ^ B + C h = ] A , Bg + ^ A , C h A + ^ B , C h = ] A + Bg , ^ A + C h

The sample coefficient of variation = CV = s/ X The sample geometric mean =

n

X1 X2 X3 fXn

The sample root-mean-square value = ^1/nh ! Xi2 When the discrete data are rearranged in increasing order and th n is odd, the median is the value of the b n + 1 l item 2 When n is even, the median is the average of the

b n l and b n + 1l items. 2 2 th

th

The mode of a set of data is the value that occurs with greatest frequency. The sample range R is the largest sample value minus the smallest sample value.

40

ENGINEERING PROBABILITY AND STATISTICS

LAWS OF PROBABILITY Property 1. General Character of Probability The probability P(E) of an event E is a real number in the range of 0 to 1. The probability of an impossible event is 0 and that of an event certain to occur is 1. Property 2. Law of Total Probability P(A + B) = P(A) + P(B) – P(A, B), where P(A + B) = the probability that either A or B occur alone or that both occur together, P(A) =

the probability that A occurs,

P(B) =

the probability that B occurs, and

P(A, B) =

the probability that both A and B occur simultaneously.

Property 3. Law of Compound or Joint Probability If neither P(A) nor P(B) is zero, P(A, B) = P(A)P(B | A) = P(B)P(A | B), where P(B | A) = the probability that B occurs given the fact that A has occurred, and

Expected Values Let X be a discrete random variable having a probability mass function

P(A | B) =

The expected value of X is defined as

the probability that A occurs given the fact that B has occurred.

If either P(A) or P(B) is zero, then P(A, B) = 0.

n = E 6 X @ = ! xk f _ xk i n

k=1

The variance of X is defined as

Bayes Theorem P _ B j Ai =

f (xk), k = 1, 2,..., n

P _ B ji P _ A B ji

v 2 = V 6 X @ = ! _ xk - ni f _ xk i n

! P _ A Bi i P _ Bi i n

i=1

where P _ A j i is the probability of event A j within the population of A

P _ B j i is the probability of event B j within the

population of B

2

k=1

Let X be a continuous random variable having a density function f(X) and let Y = g(X) be some general function. The expected value of Y is: E 6Y @ = E 7 g ] X gA = # g ^ xh f ^ xh dx 3

-3

PROBABILITY FUNCTIONS A random variable X has a probability associated with each of its possible values. The probability is termed a discrete probability if X can assume only discrete values, or X = x1, x2, x3, …, xn The discrete probability of any single event, X = xi, occurring is defined as P(xi) while the probability mass function of the random variable X is defined by f (xk) = P(X = xk), k = 1, 2, ..., n Probability Density Function If X is continuous, the probability density function, f, is defined such that P â # X # bh = # f ^ xh dx b

a

Cumulative Distribution Functions The cumulative distribution function, F, of a discrete random variable X that has a probability distribution described by P(xi) is defined as F _ xmi = ! P _ xk i = P _ X # xmi, m = 1, 2, f, n m

k=1

If X is continuous, the cumulative distribution function, F, is defined by F ^ xh = # f ^t h dt x

The mean or expected value of the random variable X is now defined as n = E 6 X @ = # xf ^ xh dx 3

-3

while the variance is given by 2 2 v 2 = V 6 X @ = E 9^ X - nh C = # ^ x - nh f ^ xh dx 3

-3

The standard deviation is given by v = V6 X @

The coefficient of variation is defined as σ/μ. Sums of Random Variables Y = a1 X1 + a2 X2 + …+ an Xn The expected value of Y is: n y = E ]Y g = a1 E ^ X1h + a2 E ^ X2h + f + an E _ Xni If the random variables are statistically independent, then the variance of Y is: v 2y = V ]Y g = a12V ^ X1h + a22V ^ X2h + f + an2V _ Xni = a12 v12 + a22 v 22 + f + an2 v 2n

Also, the standard deviation of Y is: v y = v 2y

-3

which implies that F(a) is the probability that X ≤ a.


41

Binomial Distribution P(x) is the probability that x successes will occur in n trials. If p = probability of success and q = probability of failure = 1 – p, then Pn ^ xh = C ^n, xh p x q n - x =

n! p x q n - x, x! ^ n - x h !

where x = 0, 1, 2, …, n, C(n, x) = the number of combinations, and n, p = parameters.

Normal Distribution (Gaussian Distribution) This is a unimodal distribution, the mode being x = μ, with two points of inflection (each located at a distance σ to either side of the mode). The averages of n observations tend to become normally distributed as n increases. The variate x is said to be normally distributed if its density function f (x) is given by an expression of the form f ^ xh =

1 v 2r

2 1 c x - nm e- 2 v ,

where

μ = the population mean, σ = the standard deviation of the population, and -3 # x # 3 When μ = 0 and σ2 = σ = 1, the distribution is called a standardized or unit normal distribution. Then f ^ xh =

1 e- x2/2, where -3 # x # 3. 2r

x n It is noted that Z = v follows a standardized normal distribution function. A unit normal distribution table is included at the end of this section. In the table, the following notations are utilized: F(x) = the area under the curve from –∞ to x, R(x) = the area under the curve from x to ∞, W(x) = the area under the curve between –x and x, and F(−x) = 1 − F(x) The Central Limit Theorem Let X1, X2, ..., Xn be a sequence of independent and identically distributed random variables each having mean μ and variance σ2. Then for large n, the Central Limit Theorem asserts that the sum Y = X1 + X2 + ... Xn is approximately normal. ny = n and the standard deviation vy = v n

42


t-Distribution The variate t is defined as the quotient of two independent variates x and r where x is unit normal and r is the root mean square of n other independent unit normal variates; that is, t = x/r. The following is the t-distribution with n degrees of freedom: C 7^n + 1hA /2 1 f ^t h = ^n 1h /2 C ^n/2h nr _ 2 i + 1 + t /n where –∞ ≤ t ≤ ∞.

A table at the end of this section gives the values of tα, n for values of α and n. Note that in view of the symmetry of the t-distribution, t1−α,n = –tα,n. The function for α follows: 3 a = #t f ^t h dt a, n

χ2 - Distribution If Z1, Z2, ..., Zn are independent unit normal random variables, then | 2 = Z12 + Z22 + f + Zn2 is said to have a chi-square distribution with n degrees of freedom. A table at the end of this section gives values of |a2, n for selected values of α and n. Gamma Function 3 C ^nh = #0 t n - 1e- t dt, n > 0

LINEAR REGRESSION Least Squares t , where y = at + bx t , y -intercept: at = y - bx and slope: bt = Sxy/Sxx , Sxy = ! xi yi - ^1/nh d ! xi n d ! yi n , n

n

i=1

i=1

n

i=1

Sxx = ! xi2 - ^1/nh d ! xi n , n

n

i=1

i=1

n = sample size, y = ^1/nh d ! yi n , and n

i=1

x = ^1/nh d ! xi n . n

i=1

2

Standard Error of Estimate 2 Sxx Syy - Sxy Se2 = = MSE, where Sxx ^n - 2h Syy = ! yi2 - ^1/nh d ! yi n n

n

i=1

i=1

2

Confidence Interval for a at ! ta/2, n - 2 e 1n + x o MSE Sxx 2

Confidence Interval for b bt ! ta/2, n - 2 MSE Sxx Sample Correlation Coefficient Sxy R= Sxx Syy R2 =

2 Sxy SxxSyy

HYPOTHESIS TESTING Consider an unknown parameter θ of a statistical distribution. Let the null hypothesis be H0: θ = θ0 and let the alternative hypothesis be H1: θ = θ1 Rejecting H0 when it is true is known as a type I error, while accepting H0 when it is wrong is known as a type II error. Furthermore, the probabilities of type I and type II errors are usually represented by the symbols α and β, respectively: α = probability (type I error) β = probability (type II error) The probability of a type I error is known as the level of significance of the test. Assume that the values of α and β are given. The sample size can be obtained from the following relationships. In (A) and (B), μ1 is the value assumed to be the true mean.

]Ag H0: n = n0; H1: n ! n0 b = Ue

n0 - n n -n + Za/2 o - U e 0 - Za/2 o v/ n v/ n

An approximate result is

^Za/2 + Zbh v 2 2

n -

_n1 - n0i

2

]Bg H0: n = n0; H1: n > n0 b = Ue

n0 - n + Za o v/ n

2 _ Za + Zbi v 2

n =

_n1 - n0i

2

Refer to the Hypothesis Testing table in the INDUSTRIAL ENGINEERING section of this handbook. ENGINEERING PROBABILITY AND STATISTICS

43

CONFIDENCE INTERVALS Confidence Interval for the Mean μ of a Normal Distribution

]Ag Standard deviation v is known

X - Za/2 v # n # X + Za/2 v n n ]Bg Standard deviation v is not known X - ta/2 s # n # X + ta/2 s n n where ta/2 corresponds to n - 1 degrees of freedom. Confidence Interval for the Difference Between Two Means μ1 and μ2 (A) Standard deviations σ1 and σ2 known v12 v 22 v12 v 22 # n n # Z X X + + + a / 1 2 1 2 2 n1 n2 n1 n2

X1 - X2 - Za/2

(B) Standard deviations σ1 and σ2 are not known c n1 + n1 m 8^n1 - 1h S12 + ^n2 - 1h S22 B c n1 + n1 m 8^n1 - 1h S12 + ^n2 - 1h S22B 1 2 1 2 # n1 - n2 # X1 - X2 + ta/2 X1 - X2 - ta/2 n1 + n2 - 2 n1 + n2 - 2 where ta/2 corresponds to n1 + n2 - 2 degrees of freedom.

Confidence Intervals for the Variance σ2 of a Normal Distribution ^n - 1h s 2 ^ h 2 # v 2 # 2n - 1 s 2

xa/2, n - 1

x1 - a/2, n - 1

Sample Size z=

X-n v n

2

za 2 v G n== x-n

Values of Zα/2 Confidence Interval 80% 90% 95% 96% 98% 99%

44

Zα/2 1.2816 1.6449 1.9600 2.0537 2.3263 2.5758


UNIT NORMAL DISTRIBUTION

x

f(x)

0.0 0.1 0.2 0.3 0.4

0.3989 0.3970 0.3910 0.3814 0.3683

0.5000 0.5398 0.5793 0.6179 0.6554

0.5000 0.4602 0.4207 0.3821 0.3446

1.0000 0.9203 0.8415 0.7642 0.6892

0.0000 0.0797 0.1585 0.2358 0.3108

0.5 0.6 0.7 0.8 0.9

0.3521 0.3332 0.3123 0.2897 0.2661

0.6915 0.7257 0.7580 0.7881 0.8159

0.3085 0.2743 0.2420 0.2119 0.1841

0.6171 0.5485 0.4839 0.4237 0.3681

0.3829 0.4515 0.5161 0.5763 0.6319

1.0 1.1 1.2 1.3 1.4

0.2420 0.2179 0.1942 0.1714 0.1497

0.8413 0.8643 0.8849 0.9032 0.9192

0.1587 0.1357 0.1151 0.0968 0.0808

0.3173 0.2713 0.2301 0.1936 0.1615

0.6827 0.7287 0.7699 0.8064 0.8385

1.5 1.6 1.7 1.8 1.9

0.1295 0.1109 0.0940 0.0790 0.0656

0.9332 0.9452 0.9554 0.9641 0.9713

0.0668 0.0548 0.0446 0.0359 0.0287

0.1336 0.1096 0.0891 0.0719 0.0574

0.8664 0.8904 0.9109 0.9281 0.9426

2.0 2.1 2.2 2.3 2.4

0.0540 0.0440 0.0355 0.0283 0.0224

0.9772 0.9821 0.9861 0.9893 0.9918

0.0228 0.0179 0.0139 0.0107 0.0082

0.0455 0.0357 0.0278 0.0214 0.0164

0.9545 0.9643 0.9722 0.9786 0.9836

2.5 2.6 2.7 2.8 2.9 3.0

0.0175 0.0136 0.0104 0.0079 0.0060 0.0044

0.9938 0.9953 0.9965 0.9974 0.9981 0.9987

0.0062 0.0047 0.0035 0.0026 0.0019 0.0013

0.0124 0.0093 0.0069 0.0051 0.0037 0.0027

0.9876 0.9907 0.9931 0.9949 0.9963 0.9973

0.1755 0.1031 0.0584 0.0484 0.0267 0.0145

0.9000 0.9500 0.9750 0.9800 0.9900 0.9950

0.1000 0.0500 0.0250 0.0200 0.0100 0.0050

0.2000 0.1000 0.0500 0.0400 0.0200 0.0100

0.8000 0.9000 0.9500 0.9600 0.9800 0.9900

Fractiles 1.2816 1.6449 1.9600 2.0537 2.3263 2.5758

F(x)

R(x)

2R(x)

W(x)


45

STUDENT'S t-DISTRIBUTION

α tα,n VALUES OF tα,n n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 ∞

46

0.25 1.000 0.816 0.765 0.741 0.727 0.718 0.711 0.706 0.703 0.700 0.697 0.695 0.694 0.692 0.691 0.690 0.689 0.688 0.688 0.687 0.686 0.686 0.685 0.685 0.684 0.684 0.684 0.683 0.683 0.683 0.674

0.20 1.376 1.061 0.978 0.941 0.920 0.906 0.896 0.889 0.883 0.879 0.876 0.873 0.870 0.868 0.866 0.865 0.863 0.862 0.861 0.860 0.859 0.858 0.858 0.857 0.856 0.856 0.855 0.855 0.854 0.854 0.842

0.15 1.963 1.386 1.350 1.190 1.156 1.134 1.119 1.108 1.100 1.093 1.088 1.083 1.079 1.076 1.074 1.071 1.069 1.067 1.066 1.064 1.063 1.061 1.060 1.059 1.058 1.058 1.057 1.056 1.055 1.055 1.036


0.10

0.05

0.025

0.01

0.005

n

3.078 1.886 1.638 1.533 1.476 1.440 1.415 1.397 1.383 1.372 1.363 1.356 1.350 1.345 1.341 1.337 1.333 1.330 1.328 1.325 1.323 1.321 1.319 1.318 1.316 1.315 1.314 1.313 1.311 1.310 1.282

6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.833 1.812 1.796 1.782 1.771 1.761 1.753 1.746 1.740 1.734 1.729 1.725 1.721 1.717 1.714 1.711 1.708 1.706 1.703 1.701 1.699 1.697 1.645

12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120 2.110 2.101 2.093 2.086 2.080 2.074 2.069 2.064 2.060 2.056 2.052 2.048 2.045 2.042 1.960

31.821 6.965 4.541 3.747 3.365 3.143 2.998 2.896 2.821 2.764 2.718 2.681 2.650 2.624 2.602 2.583 2.567 2.552 2.539 2.528 2.518 2.508 2.500 2.492 2.485 2.479 2.473 2.467 2.462 2.457 2.326

63.657 9.925 5.841 4.604 4.032 3.707 3.499 3.355 3.250 3.169 3.106 3.055 3.012 2.977 2.947 2.921 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797 2.787 2.779 2.771 2.763 2.756 2.750 2.576

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 ∞


47

∞

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120

Denominator df2 4 224.6 19.25 9.12 6.39 5.19 4.53 4.12 3.84 3.63 3.48 3.36 3.26 3.18 3.11 3.06 3.01 2.96 2.93 2.90 2.87 2.84 2.82 2.80 2.78 2.76 2.74 2.73 2.71 2.70 2.69 2.61 2.53 2.45 2.37

3 215.7 19.16 9.28 6.59 5.41 4.76 4.35 4.07 3.86 3.71 3.59 3.49 3.41 3.34 3.29 3.24 3.20 3.16 3.13 3.10 3.07 3.05 3.03 3.01 2.99 2.98 2.96 2.95 2.93 2.92 2.84 2.76 2.68 2.60

2

199.5 19.00 9.55 6.94 5.79 5.14 4.74 4.46 4.26 4.10 3.98 3.89 3.81 3.74 3.68 3.63 3.59 3.55 3.52 3.49 3.47 3.44 3.42 3.40 3.39 3.37 3.35 3.34 3.33 3.32 3.23 3.15 3.07 3.00

1

161.4 18.51 10.13 7.71 6.61 5.99 5.59 5.32 5.12 4.96 4.84 4.75 4.67 4.60 4.54 4.49 4.45 4.41 4.38 4.35 4.32 4.30 4.28 4.26 4.24 4.23 4.21 4.20 4.18 4.17 4.08 4.00 3.92 3.84

For a particular combination of numerator and denominator degrees of freedom, entry represents the critical values of F corresponding to a specified upper tail area (α).

230.2 19.30 9.01 6.26 5.05 4.39 3.97 3.69 3.48 3.33 3.20 3.11 3.03 2.96 2.90 2.85 2.81 2.77 2.74 2.71 2.68 2.66 2.64 2.62 2.60 2.59 2.57 2.56 2.55 2.53 2.45 2.37 2.29 2.21

5

6 234.0 19.33 8.94 6.16 4.95 4.28 3.87 3.58 3.37 3.22 3.09 3.00 2.92 2.85 2.79 2.74 2.70 2.66 2.63 2.60 2.57 2.55 2.53 2.51 2.49 2.47 2.46 2.45 2.43 2.42 2.34 2.25 2.17 2.10

0

8 238.9 19.37 8.85 6.04 4.82 4.15 3.73 3.44 3.23 3.07 2.95 2.85 2.77 2.70 2.64 2.59 2.55 2.51 2.48 2.45 2.42 2.40 2.37 2.36 2.34 2.32 2.31 2.29 2.28 2.27 2.18 2.10 2.02 1.94

7 236.8 19.35 8.89 6.09 4.88 4.21 3.79 3.50 3.29 3.14 3.01 2.91 2.83 2.76 2.71 2.66 2.61 2.58 2.54 2.51 2.49 2.46 2.44 2.42 2.40 2.39 2.37 2.36 2.35 2.33 2.25 2.17 2.09 2.01 240.5 19.38 8.81 6.00 4.77 4.10 3.68 3.39 3.18 3.02 2.90 2.80 2.71 2.65 2.59 2.54 2.49 2.46 2.42 2.39 2.37 2.34 2.32 2.30 2.28 2.27 2.25 2.24 2.22 2.21 2.12 2.04 1.96 1.88

9 241.9 19.40 8.79 5.96 4.74 4.06 3.64 3.35 3.14 2.98 2.85 2.75 2.67 2.60 2.54 2.49 2.45 2.41 2.38 2.35 2.32 2.30 2.27 2.25 2.24 2.22 2.20 2.19 2.18 2.16 2.08 1.99 1.91 1.83

10

15 245.9 19.43 8.70 5.86 4.62 3.94 3.51 3.22 3.01 2.85 2.72 2.62 2.53 2.46 2.40 2.35 2.31 2.27 2.23 2.20 2.18 2.15 2.13 2.11 2.09 2.07 2.06 2.04 2.03 2.01 1.92 1.84 1.75 1.67

12 243.9 19.41 8.74 5.91 4.68 4.00 3.57 3.28 3.07 2.91 2.79 2.69 2.60 2.53 2.48 2.42 2.38 2.34 2.31 2.28 2.25 2.23 2.20 2.18 2.16 2.15 2.13 2.12 2.10 2.09 2.00 1.92 1.83 1.75

Numerator df1

F(α, df1, df2)

α = 0.05

CRITICAL VALUES OF THE F DISTRIBUTION – TABLE

20 248.0 19.45 8.66 5.80 4.56 3.87 3.44 3.15 2.94 2.77 2.65 2.54 2.46 2.39 2.33 2.28 2.23 2.19 2.16 2.12 2.10 2.07 2.05 2.03 2.01 1.99 1.97 1.96 1.94 1.93 1.84 1.75 1.66 1.57

F

249.1 19.45 8.64 5.77 4.53 3.84 3.41 3.12 2.90 2.74 2.61 2.51 2.42 2.35 2.29 2.24 2.19 2.15 2.11 2.08 2.05 2.03 2.01 1.98 1.96 1.95 1.93 1.91 1.90 1.89 1.79 1.70 1.61 1.52

24

40 251.1 19.47 8.59 5.72 4.46 3.77 3.34 3.04 2.83 2.66 2.53 2.43 2.34 2.27 2.20 2.15 2.10 2.06 2.03 1.99 1.96 1.94 1.91 1.89 1.87 1.85 1.84 1.82 1.81 1.79 1.69 1.59 1.50 1.39

30 250.1 19.46 8.62 5.75 4.50 3.81 3.38 3.08 2.86 2.70 2.57 2.47 2.38 2.31 2.25 2.19 2.15 2.11 2.07 2.04 2.01 1.98 1.96 1.94 1.92 1.90 1.88 1.87 1.85 1.84 1.74 1.65 1.55 1.46

252.2 19.48 8.57 5.69 4.43 3.74 3.30 3.01 2.79 2.62 2.49 2.38 2.30 2.22 2.16 2.11 2.06 2.02 1.98 1.95 1.92 1.89 1.86 1.84 1.82 1.80 1.79 1.77 1.75 1.74 1.64 1.53 1.43 1.32

60

253.3 19.49 8.55 5.66 4.40 3.70 3.27 2.97 2.75 2.58 2.45 2.34 2.25 2.18 2.11 2.06 2.01 1.97 1.93 1.90 1.87 1.84 1.81 1.79 1.77 1.75 1.73 1.71 1.70 1.68 1.58 1.47 1.35 1.22

120

254.3 19.50 8.53 5.63 4.36 3.67 3.23 2.93 2.71 2.54 2.40 2.30 2.21 2.13 2.07 2.01 1.96 1.92 1.88 1.84 1.81 1.78 1.76 1.73 1.71 1.69 1.67 1.65 1.64 1.62 1.51 1.39 1.25 1.00

∞

48


2 .995

0.0000393 0.0100251 0.0717212 0.206990 0.411740 0.675727 0.989265 1.344419 1.734926 2.15585 2.60321 3.07382 3.56503 4.07468 4.60094 5.14224 5.69724 6.26481 6.84398 7.43386 8.03366 8.64272 9.26042 9.88623 10.5197 11.1603 11.8076 12.4613 13.1211 13.7867 20.7065 27.9907 35.5346 43.2752 51.1720 59.1963 67.3276

X

X α,n

2

X

2 .990

0.0001571 0.0201007 0.114832 0.297110 0.554300 0.872085 1.239043 1.646482 2.087912 2.55821 3.05347 3.57056 4.10691 4.66043 5.22935 5.81221 6.40776 7.01491 7.63273 8.26040 8.89720 9.54249 10.19567 10.8564 11.5240 12.1981 12.8786 13.5648 14.2565 14.9535 22.1643 29.7067 37.4848 45.4418 53.5400 61.7541 70.0648

α

X

2

2 .975

0.0009821 0.0506356 0.215795 0.484419 0.831211 1.237347 1.68987 2.17973 2.70039 3.24697 3.81575 4.40379 5.00874 5.62872 6.26214 6.90766 7.56418 8.23075 8.90655 9.59083 10.28293 10.9823 11.6885 12.4011 13.1197 13.8439 14.5733 15.3079 16.0471 16.7908 24.4331 32.3574 40.4817 48.7576 57.1532 65.6466 74.2219

X

2 .950

0.0039321 0.102587 0.351846 0.710721 1.145476 1.63539 2.16735 2.73264 3.32511 3.94030 4.57481 5.22603 5.89186 6.57063 7.26094 7.96164 8.67176 9.39046 10.1170 10.8508 11.5913 12.3380 13.0905 13.8484 14.6114 15.3791 16.1513 16.9279 17.7083 18.4926 26.5093 34.7642 43.1879 51.7393 60.3915 69.1260 77.9295

X

2 .900

0.0157908 0.210720 0.584375 1.063623 1.61031 2.20413 2.83311 3.48954 4.16816 4.86518 5.57779 6.30380 7.04150 7.78953 8.54675 9.31223 10.0852 10.8649 11.6509 12.4426 13.2396 14.0415 14.8479 15.6587 16.4734 17.2919 18.1138 18.9392 19.7677 20.5992 29.0505 37.6886 46.4589 55.3290 64.2778 73.2912 82.3581

X

2 .100

2.70554 4.60517 6.25139 7.77944 9.23635 10.6446 12.0170 13.3616 14.6837 15.9871 17.2750 18.5494 19.8119 21.0642 22.3072 23.5418 24.7690 25.9894 27.2036 28.4120 29.6151 30.8133 32.0069 33.1963 34.3816 35.5631 36.7412 37.9159 39.0875 40.2560 51.8050 63.1671 74.3970 85.5271 96.5782 107.565 118.498

X

Source: Thompson, C. M., “Tables of the Percentage Points of the X2–Distribution,” Biometrika, ©1941, 32, 188-189. Reproduced by permission of the Biometrika Trustees.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 50 60 70 80 90 100

Degrees of Freedom

0

f (X 2 )

CRITICAL VALUES OF X 2 DISTRIBUTION

2 .050

3.84146 5.99147 7.81473 9.48773 11.0705 12.5916 14.0671 15.5073 16.9190 18.3070 19.6751 21.0261 22.3621 23.6848 24.9958 26.2962 27.5871 28.8693 30.1435 31.4104 32.6705 33.9244 35.1725 36.4151 37.6525 38.8852 40.1133 41.3372 42.5569 43.7729 55.7585 67.5048 79.0819 90.5312 101.879 113.145 124.342

X

2 .025

5.02389 7.37776 9.34840 11.1433 12.8325 14.4494 16.0128 17.5346 19.0228 20.4831 21.9200 23.3367 24.7356 26.1190 27.4884 28.8454 30.1910 31.5264 32.8523 34.1696 35.4789 36.7807 38.0757 39.3641 40.6465 41.9232 43.1944 44.4607 45.7222 46.9792 59.3417 71.4202 83.2976 95.0231 106.629 118.136 129.561

X

2

.010

6.63490 9.21034 11.3449 13.2767 15.0863 16.8119 18.4753 20.0902 21.6660 23.2093 24.7250 26.2170 27.6883 29.1413 30.5779 31.9999 33.4087 34.8053 36.1908 37.5662 38.9321 40.2894 41.6384 42.9798 44.3141 45.6417 46.9630 48.2782 49.5879 50.8922 63.6907 76.1539 88.3794 100.425 112.329 124.116 135.807

X

2

.005

7.87944 10.5966 12.8381 14.8602 16.7496 18.5476 20.2777 21.9550 23.5893 25.1882 26.7569 28.2995 29.8194 31.3193 32.8013 34.2672 35.7185 37.1564 38.5822 39.9968 41.4010 42.7956 44.1813 45.5585 46.9278 48.2899 49.6449 50.9933 52.3356 53.6720 66.7659 79.4900 91.9517 104.215 116.321 128.299 140.169

X

critical values of x2 distribution

STATICS FORCE A force is a vector quantity. It is defined when its (1) magnitude, (2) point of application, and (3) direction are known. RESULTANT (TWO DIMENSIONS) The resultant, F, of n forces with components Fx,i and Fy,i has the magnitude of 1 2

F = >d ! Fx, i n + d ! Fy, i n H 2

n

2

n

i=1

i=1

The resultant direction with respect to the x-axis using fourquadrant angle functions is i = arctan e ! Fy, i n

i=1

! Fx, i o n

i=1

The vector form of a force is F = Fx i + Fy j

RESOLUTION OF A FORCE Fx = F cos θx; Fy = F cos θy; Fz = F cos θz cos θx = Fx /F; cos θy = Fy /F; cos θz = Fz /F Separating a force into components when the geometry of force is known and R = x 2 + y 2 + z 2 Fx = (x/R)F;

Fy = (y/R)F;

Fz = (z/R)F

MOMENTS (COUPLES) A system of two forces that are equal in magnitude, opposite in direction, and parallel to each other is called a couple. A moment M is defined as the cross product of the radius vector r and the force F from a point to the line of action of the force. M = r × F;

Mx = yFz – zFy, My = zFx – xFz, and Mz = xFy – yFx.

SYSTEMS OF FORCES F = Σ Fn M = Σ (rn × Fn) Equilibrium Requirements Σ Fn = 0 Σ Mn = 0

CENTROIDS OF MASSES, AREAS, LENGTHS, AND VOLUMES Formulas for centroids, moments of inertia, and first moment of areas are presented in the MATHEMATICS section for continuous functions. The following discrete formulas are for defined regular masses, areas, lengths, and volumes: rc = Σ mnrn /Σ mn, where mn = the mass of each particle making up the system, rn = the radius vector to each particle from a selected reference point, and rc = the radius vector to the center of the total mass from the selected reference point. The moment of area (Ma) is defined as May = Σ xnan Max = Σ ynan Maz = Σ znan The centroid of area is defined as xac = May /A yac = Max /A with respect to center of zac = Maz /A the coordinate system where A = Σ an The centroid of a line is defined as xlc = (Σ xnln)/L, where L = Σ ln ylc = (Σ ynln)/L zlc = (Σ znln)/L The centroid of volume is defined as xvc = (Σ xnvn)/V, where V = Σ vn yvc = (Σ ynvn)/V zvc = (Σ znvn)/V

MOMENT OF INERTIA The moment of inertia, or the second moment of area, is defined as Iy = ∫x2 dA Ix = ∫y2 dA The polar moment of inertia J of an area about a point is equal to the sum of the moments of inertia of the area about any two perpendicular axes in the area and passing through the same point. Iz = J = Iy + Ix = ∫(x2 + y2) dA = rp2A, where rp = the radius of gyration (see the DYNAMICS section and the next page of this section).

STATICS

49

Moment of Inertia Parallel Axis Theorem The moment of inertia of an area about any axis is defined as the moment of inertia of the area about a parallel centroidal axis plus a term equal to the area multiplied by the square of the perpendicular distance d from the centroidal axis to the axis in question. I lx = Ixc + dx2 A I ly = Iyc + d 2y A, where dx, dy

= distance between the two axes in question,

Ixc , Iyc = the moment of inertia about the centroidal axis, and I lx, I ly = the moment of inertia about the new axis. Radius of Gyration The radius of gyration rp, rx, ry is the distance from a reference axis at which all of the area can be considered to be concentrated to produce the moment of inertia. rx =

Ix A; ry =

Iy A; rp =

J A

Product of Inertia The product of inertia (Ixy, etc.) is defined as: Ixy = ∫xydA, with respect to the xy-coordinate system, Ixz = ∫xzdA, with respect to the xz-coordinate system, and Iyz = ∫yzdA, with respect to the yz-coordinate system. The parallel-axis theorem also applies: I'xy = Ixc yc + dxdy A for the xy-coordinate system, etc. where dx = x-axis distance between the two axes in question, and dy = y-axis distance between the two axes in question.

FRICTION The largest frictional force is called the limiting friction. Any further increase in applied forces will cause motion. F ≤ μsN, where

BELT FRICTION F1 = F2 eμθ, where F1 = force being applied in the direction of impending motion, F2 = force applied to resist impending motion, μ = coefficient of static friction, and θ = the total angle of contact between the surfaces expressed in radians. STATICALLY DETERMINATE TRUSS Plane Truss A plane truss is a rigid framework satisfying the following conditions: 1. The members of the truss lie in the same plane. 2. The members are connected at their ends by frictionless pins. 3. All of the external loads lie in the plane of the truss and are applied at the joints only. 4. The truss reactions and member forces can be determined using the equations of equilibrium. Σ F = 0; Σ M = 0 5. A truss is statically indeterminate if the reactions and member forces cannot be solved with the equations of equilibrium. Plane Truss: Method of Joints The method consists of solving for the forces in the members by writing the two equilibrium equations for each joint of the truss. Σ FV = 0 and Σ FH = 0, where FH = horizontal forces and member components and FV = vertical forces and member components. Plane Truss: Method of Sections The method consists of drawing a free-body diagram of a portion of the truss in such a way that the unknown truss member force is exposed as an external force.

F = friction force, μs = coefficient of static friction, and N = normal force between surfaces in contact.

SCREW THREAD (also see MECHANICAL ENGINEERING section) For a screw-jack, square thread, M = Pr tan (α ± φ), where + is for screw tightening, – is for screw loosening, M = external moment applied to axis of screw, P = load on jack applied along and on the line of the axis, r = the mean thread radius, α = the pitch angle of the thread, and μ = tan φ = the appropriate coefficient of friction. 50

STATICS

CONCURRENT FORCES A concurrent-force system is one in which the lines of action of the applied forces all meet at one point. A two-force body in static equilibrium has two applied forces that are equal in magnitude, opposite in direction, and collinear. A three-force body in static equilibrium has three applied forces whose lines of action meet at a point. As a consequence, if the direction and magnitude of two of the three forces are known, the direction and magnitude of the third can be determined.

STATICS

51

C

b

C

x

x

x

b

x

h

x

h

h

a

b

h

a

x

h

C

b

C

a

C

b

b

C = b3h/4

Iy

= b3h/12

= bh3/12

[ (

[ (

)

(

)

2

(

]3

− a 2b 2 sinθ cosθ 6

2

I y = ab sinθ (b + a cosθ )

yc = (a sin θ)/2

[

Ix

3

xc = (b + a cos θ)/2

3

I yc

2

I xc = a 3b sin 3θ 12

( ) = [ab sinθ (b + a cos θ )] 12 = (a b sin θ ) 3 2

h 3 (3a + b ) 12

Ix =

(

h 3 a 2 + 4ab + b 2 36(a + b )

I xc =

A = ab sin θ

A = h(a + b ) 2 h(2a + b ) yc = 3(a + b )

)]

J = bh b 2 + h 2 12

[ (

)

) 18 )

)

rx2 ry2

(

)

= (b + a cosθ )2 3 − (ab cosθ ) 6

= (a sinθ ) 3

2

ry2c = b 2 + a 2 cos 2 θ 12

rx2c = (a sinθ )2 12

(

h 2 a 2 + 4ab + b 2 18(a + b ) 2 ( h a + b) 3 rx2 = 6(a + b ) rx2c =

rp2 = b 2 + h 2 12

(

rx2 = h 2 3 ry2 = b 2 3

I x = bh 3 3 I y = b3h 3

xc = b/2

(

ry2c = b 2 12

yc = h/2

2

=h 6 = b 2 + ab + a 2 6

2

= b − ab + a

2

I yc = b 3 h 12

)] 12

)] 36 ry2c rx2 ry2

A = bh

2

2

rx2c = h 2 18

= Abh 12 = b 2 h 2 24

[

]

= bh 2 (2a + b ) 24

(

)

= Abh 4 = b 2 h 2 4

I xc yc = a 3b sin 2 θ cos θ 12

I xy

]

= [Ah(2a + b )] 12

I xc y c = 0

I xy

[

= bh 2 (2a − b ) 72

I xc yc = [Ah(2a − b )] 36

I xy

rx2 = h 2 6 ry2 = b 2 6

= Abh 4 = b 2 h 2 8

I xc yc = − Abh 36 = − b 2 h 2 72

Ixy

Ixc yc = Abh 36 = b 2 h 2 72

Product of Inertia

ry2c = b 2 18

rx2c = h 2 18

rx2 = h 2 6 ry2 = b 2 2

rx2c = h 2 12

I y = bh b + ab + a

2

I x = bh 12

3

Iyc = bh b − ab + a

2

I xc = bh 3 36

Iy

Ix

I yc = b 3h/36

Ix c = bh 3 /36

= bh3/12

ry2c = b 2 18

I yc = b 3h/36

Ix

rx2c = h 2 18

(Radius of Gyration)2

Ix c = bh 3 /36

Area Moment of Inertia

I xc = b h 3 12

yc = h/3

xc = (a + b)/3

A = bh/2

yc = h/3

xc = b/3

A = bh/2

yc = h/3

xc = 2b/3

A = bh/2

Area & Centroid

Housner, George W. & Donald E. Hudson, Applied Mechanics Dynamics, D. Van Nostrand Company, Inc., Princeton, NJ, 1959. Table reprinted by permission of G.W. Housner & D.E. Hudson.

y

y

y

y

y

y

Figure

52

STATICS

y

y

a

a

a

b

a

C

2a

C

C

C

C

x

x

x

x

x

2

[

2a sin 3θ 3 θ − sinθ cosθ

yc = 0

xc =

A = a2 θ −

sin 2θ 2

A = a 2θ 2a sinθ xc = 3 θ yc = 0

yc = 4a/(3π)

xc = a

A = πa /2

yc = a

xc = a

A = π (a2 – b2)

(

)

Iy =

Ix =

4

2sin 3 θ cosθ Aa 2 1+ 4 θ − sin θ cosθ

]

2sin 3θ cosθ Aa 2 1− 4 3θ − 3sinθ cosθ

[ [

Iy = a4(θ + sinθ cosθ)/4

Ix = a (θ – sinθ cosθ)/4

4

I x = πa 4 8 I y = 5π a 4 8

I yc

(

)

a 4 9π 2 − 64 72π = πa 4 8

I xc =

(

5πa πb Ix = Iy = − πa 2b 2 − 4 4 J = π a4 − b4 2

4

I xc = I y c = π a 4 − b 4 4

]

) +b ) 4

=

)

2

)

2

ry2 =

rx2 =

2sin 3θ cosθ a2 1+ 4 θ − sinθ cosθ

[ [

]

2sin 3θ cosθ a2 1− 4 3θ − 3sinθ cosθ

a 2 (θ − sinθ cosθ ) θ 4 a 2 (θ + sinθ cosθ ) 2 ry = θ 4 rx2 =

ry2 = 5a 2 4

rx2 = a 2 4

(

a 2 9π 2 − 64 36π 2 2 2 ryc = a 4 rx2c =

(

ry2

rp2 = a 2 + b 2 2

rx2

( = (5a

rx2c = ry2c = a 2 + b 2 4

=a 2

rp2

J = πa 4 2

yc = a 2

rx2 = ry2 = 5a 2 4

I x = I y = 5π a 4 4

xc = a

rx2c = ry2c = a 2 4

]


I xc = I y c = π a 4 4

)


A = πa2

]

Area & Centroid


y

y

y

Figure

(

I xc y c = 0 I xy = 0

I xc y c = 0 I xy = 0

I xy = 2a 4 3

I xc y c = 0

= πa 2 a 2 − b 2

I xy = Aa 2

I xc y c = 0

I xy = Aa 2

I xc y c = 0

)

Product of Inertia

STATICS

53

b

C

x

h

y = (h/b 1/n)x1/n

b

x

b

b

b

C

y = (h/b n)xn

a

C

a

C

x

h

x

n +1 b n+2

n +1 h 2(n + 2)

n +1 b 2n + 1

n bh n +1

yc =

xc =

A=

h n +1 yc = 2 2n + 1

xc =

A = bh (n + 1)

yc = 3b/8

xc = 3a/5

A = 2ab/3

yc = 0

xc = 3a/5

A = 4ab/3

Area & Centroid

hb 3 n+3

n Ix = bh 3 3(n + 3) n Iy = b3h 3n + 1

Iy =

bh 3 Ix = 3(3n + 1)

Iy = 2ba /7

3

Ix = 2ab3/15

I y = 4 a 3b 7

I yc = 16a b 175

3

I xc = I x = 4ab 3 15


= 12a 175

2

ry2 =

rx2 =

ry2 =

rx2 =

n +1 2 b 3n + 1

n +1 2 h 3(n + 1)

n +1 2 b n+3

h 2 (n + 1) 3(3n + 1)

ry2 = 3a 2 7

rx2 = b 2 5

ry2 = 3a 2 7

ry2c

rx2c = rx2 = b 2 5



y

y

y

y

Figure

Ixy = Aab/4 = a2b2

Ixy = 0

I xc y c = 0

Product of Inertia

DYNAMICS KINEMATICS Kinematics is the study of motion without consideration of the mass of, or the forces acting on, the system. For particle motion, let r(t) be the position vector of the particle in an inertial reference frame. The velocity and acceleration of the particle are respectively defined as v = dr/dt a = dv/dt, where v = the instantaneous velocity, a = the instantaneous acceleration, and t = time Cartesian Coordinates r = xi + yj + zk v = xo i + yo j + zo k a = xp i + yp j + zp k, where xo = dx/dt = vx, etc.

er r θ

s x

r = rer v = r~et

a = _- r~ 2i er + raet, where r = the radius of the circle, and

Radial and Transverse Components for Planar Motion y

eθ er

i = the angle between the x and er axes The magnitudes of the angular velocity and acceleration, respectively, are defined as ~ = io , and a = ~o = ip

r θ x

Unit vectors eθ and er are, respectively, normal to and collinear with the position vector r. Thus: r = rer v = ro er + rio ei a = _ rp - rio 2i er + ^rip + 2ro io h ei, where r = the radial distance i = the angle between the x axis and er, ro = dr/dt, etc. rp = d 2r/dt 2, etc.

et

Here the vector quantities are defined as

xp = d 2x/dt 2 = ax, etc.

PATH

Plane Circular Motion A special case of transverse and radial components is for constant radius rotation about the origin, or plane circular motion. y

Arc length, tangential velocity, and tangential acceleration, respectively, are s = ri, vt = r~ at = ra The normal acceleration is given by an = r~ 2 Normal and Tangential Components y et en r

PATH

54

DYNAMICS

x

Unit vectors et and en are, respectively, tangent and normal to the path with en pointing to the center of curvature. Thus v = v ^t h et

a = a ^t h et + ` vt2/tj en, where

Non-constant Acceleration When non-constant acceleration, a(t), is considered, the equations for the velocity and displacement may be obtained from v ^t h = # a ^xh dx + vt0 t

t = instantaneous radius of curvature

t0

s ^t h = # v ^xh dx + st0 t

Constant Acceleration The equations for the velocity and displacement when acceleration is a constant are given as a(t) = a0 v(t) = a0 (t – t0) + v0 s(t) = a0 (t – t0)2/2 + v0 (t – t0) + s0, where s = distance along the line of travel s0 = displacement at time t0 v = velocity along the direction of travel v0 = velocity at time t0 a0 = constant acceleration t = time, and t0 = some initial time

t0

For variable angular acceleration ~ ^t h = # a ^xh dx + ~t0 t

t0

i ^t h = # ~ ^xh dx + it0 t

t0

Projectile Motion

y

For a free-falling body, a0 = g (downward). An additional equation for velocity as a function of position may be written as v 2 = v02 + 2a0 _ s - s0i For constant angular acceleration, the equations for angular velocity and displacement are a ^t h = a0

~ ^t h = a0 _t - t0i + ~0

v0 θ

x The equations for common projectile motion may be obtained from the constant acceleration equations as ax = 0 vx = v0 cos(θ) = v0 cos(θ)t + x0

i ^t h = a0 _t - t0i /2 + ~0 _t - t0i + i0, where

x

angular displacement

θ0 =

vy = –gt + v0 sin(θ)

angular displacement at time t0

y

ω

angular velocity

2

θ

= =

ω0 =

angular velocity at time t0

α0 =

constant angular acceleration

t

=

time, and

t0

=

some initial time

An additional equation for angular velocity as a function of angular position may be written as

ay = –g = –gt2/2 + v0 sin(θ)t + y0

CONCEPT OF WEIGHT W = mg, where W = weight, N (lbf), m = mass, kg (lbf-sec2/ft), and g

= local acceleration of gravity, m/sec2 (ft/sec2)

~ 2 = ~02 + 2a0 _i - i0i

DYNAMICS

55

KINETICS Newton’s second law for a particle is ΣF = d(mv)/dt, where ΣF = the sum of the applied forces acting on the particle, m = the mass of the particle v

= the velocity of the particle

For constant mass, ΣF = m dv/dt = ma One-Dimensional Motion of a Particle (Constant Mass) When motion only exists in a single dimension then, without loss of generality, it may be assumed to be in the x direction, and ax = Fx /m, where Fx = the resultant of the applied forces which in general can depend on t, x, and vx. If Fx only depends on t, then ax ^t h = Fx ^t h /m,

The term on the left side of the equation is the linear momentum of a system of particles at time t2. The first term on the right side of the equation is the linear momentum of a system of particles at time t1. The second term on the right side of the equation is the impulse of the force F from time t1 to t2. It should be noted that the above equation is a vector equation. Component scalar equations may be obtained by considering the momentum and force in a set of orthogonal directions. Angular Momentum or Moment of Momentum The angular momentum or the moment of momentum about point 0 of a particle is defined as H0 = r # mv, or H0 = I0 ~ Taking the time derivative of the above, the equation of motion may be written as Ho 0 = d _ I0 ~i /dt = M, where M is the moment applied to the particle. Now by integrating and summing over a system of any number of particles, this may be expanded to

vx ^t h = # ax ^xh dx + vxt 0 , t

t0

R ^H0iht 2 = R ^H0iht 1 + R # M0idt t2

x ^t h = # vx ^xh dx + xt 0 t

t1

t0

If the force is constant (i.e. independent of time, displacement, and velocity) then ax = Fx /m,

vx = ax _t - t0i + vxt 0,

x = ax _t - t0i /2 + vxt 0 _t - t0i + xt 0

The term on the left side of the equation is the angular momentum of a system of particles at time t2. The first term on the right side of the equation is the angular momentum of a system of particles at time t1. The second term on the right side of the equation is the angular impulse of the moment M from time t1 to t2.

2

Normal and Tangential Kinetics for Planar Problems When working with normal and tangential directions, the scalar equations may be written as RFt = mat = mdvt /dt and RFn = man = m ` vt2/tj

Impulse and Momentum Linear Assuming constant mass, the equation of motion of a particle may be written as mdv/dt = F mdv =Fdt For a system of particles, by integrating and summing over the number of particles, this may be expanded to Rmi ^viht 2 = Rmi ^viht1 + R # Fi dt t2

t1

56

DYNAMICS

Work and Energy Work W is defined as W = ∫F • dr (For particle flow, see FLUID MECHANICS section.) Kinetic Energy The kinetic energy of a particle is the work done by an external agent in accelerating the particle from rest to a velocity v. Thus T = mv2/2 In changing the velocity from v1 to v2, the change in kinetic energy is T2 - T1 = m ` v22 - v12j /2

Potential Energy The work done by an external agent in the presence of a conservative field is termed the change in potential energy. Potential Energy in Gravity Field U = mgh, where h = the elevation above some specified datum. Elastic Potential Energy For a linear elastic spring with modulus, stiffness, or spring constant, the force in the spring is Fs = k x, where x = the change in length of the spring from the undeformed length of the spring. The potential energy stored in the spring when compressed or extended by an amount x is U = k x2/2 In changing the deformation in the spring from position x1 to x2, the change in the potential energy stored in the spring is U2 - U1 = k ` x22 - x12j /2 Principle of Work and Energy If Ti and Ui are, respectively, the kinetic and potential energy of a particle at state i, then for conservative systems (no energy dissipation or gain), the law of conservation of energy is T2 +U2 = T1 +U1 If nonconservative forces are present, then the work done by these forces must be accounted for. Hence T2 +U2 = T1 +U1 +W1→2 , where W1→2 = the work done by the nonconservative forces in moving between state 1 and state 2. Care must be exercised during computations to correctly compute the algebraic sign of the work term. If the forces serve to increase the energy of the system, W1→2 is positive. If the forces, such as friction, serve to dissipate energy, W1→2 is negative. Impact During an impact, momentum is conserved while energy may or may not be conserved. For direct central impact with no external forces m1v 1 + m2v 2 = m1v l1 + m2v l2, where m1, m2 = the masses of the two bodies, v1, v2

= the velocities of the bodies just before impact, and

v l1, v l2

= the velocities of the bodies just after impact.

For impacts, the relative velocity expression is e=

^v l2hn - ^v l1hn , where ^v1hn - ^v2hn

e = coefficient of restitution, (vi)n = the velocity normal to the plane of impact just before impact, and _ v li in = the velocity normal to the plane of impact just after impact The value of e is such that 0 ≤ e ≤ 1, with limiting values e = 1, perfectly elastic (energy conserved), and e = 0, perfectly plastic (no rebound) Knowing the value of e, the velocities after the impact are given as m2 ^v2hn ^1 + eh + ^m1 - em2h ^v1hn ^v l1hn = m1 + m2

^v l2hn =

m1 ^v1hn ^1 + eh - êm1 - m2h ^v2hn m1 + m2

Friction The Laws of Friction are 1. The total friction force F that can be developed is independent of the magnitude of the area of contact. 2. The total friction force F that can be developed is proportional to the normal force N. 3. For low velocities of sliding, the total frictional force that can be developed is practically independent of the velocity, although experiments show that the force F necessary to initiate slip is greater than that necessary to maintain the motion. The formula expressing the Laws of Friction is F ≤ μN, where μ = the coefficient of friction. In general F < μs N, no slip occuring, F = μs N, at the point of impending slip, and F = μk N, when slip is occuring. Here, μs = the coefficient of static friction, and μk = the coefficient of kinetic friction. The coefficient of kinetic friction is often approximated as 75% of the coefficient of static friction.

DYNAMICS

57

Mass Moment of Inertia The definitions for the mass moments of inertia are Ix = # ` y 2 + z 2j dm,

Iy = # _ x 2 + z 2i dm, and Iz = # ` x 2 + y 2j dm

A table listing moment of inertia formulas for some standard shapes is at the end of this section. Parallel-Axis Theorem The mass moments of inertia may be calculated about any axis through the application of the above definitions. However, once the moments of inertia have been determined about an axis passing through a body’s mass center, it may be transformed to another parallel axis. The transformation equation is Inew = Ic + md 2 , where Inew = the mass moment of inertia about any specified axis Ic

= the mass moment of inertia about an axis that is parallel to the above specified axis but passes through the body’s mass center

The figure shows a fourbar slider-crank. Link 2 (the crank) rotates about the fixed center, O2. Link 3 couples the crank to the slider (link 4), which slides against ground (link 1). Using the definition of an instant center (IC), we see that the pins at O2, A, and B are ICs that are designated I12, I23, and I34. The easily observable IC is I14, which is located at infinity with its direction perpendicular to the interface between links 1 and 4 (the direction of sliding). To locate the remaining two ICs (for a fourbar) we must make use of Kennedy’s rule. Kennedy’s Rule: When three bodies move relative to one another they have three instantaneous centers, all of which lie on the same straight line. To apply this rule to the slider-crank mechanism, consider links 1, 2, and 3 whose ICs are I12, I23, and I13, all of which lie on a straight line. Consider also links 1, 3, and 4 whose ICs are I13, I34, and I14, all of which lie on a straight line. Extending the line through I12 and I23 and the line through I34 and I14 to their intersection locates I13, which is common to the two groups of links that were considered. I14 ∞ I13

I14 ∞

m = the mass of the body d

= the normal distance from the body’s mass center to the above-specified axis

Radius of Gyration The radius of gyration is defined as r=

I12

I23 2

Kinematics Instantaneous Center of Rotation (Instant Centers) An instantaneous center of rotation (instant center) is a point, common to two bodies, at which each has the same velocity (magnitude and direction) at a given instant. It is also a point on one body about which another body rotates, instantaneously. A B

3

2

θ2

O2

4 1 GROUND I 14 ∞ I 23

I12

DYNAMICS

3

I34 4

1 GROUND

I m

PLANE MOTION OF A RIGID BODY

58

I24

I 34

Similarly, if body groups 1, 2, 4 and 2, 3, 4 are considered, a line drawn through known ICs I12 and I14 to the intersection of a line drawn through known ICs I23 and I34 locates I24. The number of ICs, c, for a given mechanism is related to the number of links, n, by ^ h c=n n-1 2

Relative Motion The equations for the relative position, velocity, and acceleration may be written as

RFx = maxc RFy = mayc RMzc = Izc a, where

Translating Axis y

Y

A rrel x

B

rB

Without loss of generality, the body may be assumed to be in the x-y plane. The scalar equations of motion may then be written as

zc indicates the z axis passing through the body’s mass center, axc and ayc are the acceleration of the body’s mass center in the x and y directions, respectively, and α is the angular acceleration of the body about the z axis.

X

rA = rB + rrel vA = vB + ω × rrel aA = aB + α × rrel + ω × (ω × rrel) where, ω and α are, respectively, the angular velocity and angular acceleration of the relative position vector rrel. Rotating Axis Y

A y

rB

rrel

x

Rotation about an Arbitrary Fixed Axis For rotation about some arbitrary fixed axis q RMq = Iq a If the applied moment acting about the fixed axis is constant then integrating with respect to time, from t = 0 yields α ω θ

= = =

Mq /Iq ω0 + α t θ0 + ω0 t + α t2/2

where ω0 and θ0 are the values of angular velocity and angular displacement at time t = 0 , respectively.

B

X

rA = rB + rrel vA = vB + ω × rrel + vrel aA = aB + α × rrel + ω × (ω × rrel) + 2ω × vrel + arel where, ω and α are, respectively, the total angular velocity and acceleration of the relative position vector rrel. Rigid Body Rotation For rigid body rotation θ ω = dθ/dt, α = dω/dt, and αdθ = ωdω Kinetics In general, Newton’s second laws for a rigid body, with constant mass and mass moment of inertia, in plane motion may be written in vector form as

The change in kinetic energy is the work done in accelerating the rigid body from ω0 to ω i

Iq ~ 2/2 = Iq ~02 /2 + # Mqdi i0

Kinetic Energy In general the kinetic energy for a rigid body may be written as T = mv 2/2 + Ic ~ 2/2 For motion in the xy plane this reduces to 2 2 T = m `vcx j /2 + Ic ~ 2z /2 + vcy

For motion about an instant center, T = IICω2/2

RF = mac RMc = Ic a RMp = Ic a + t pc # mac, where F are forces and ac is the acceleration of the body’s mass center both in the plane of motion, Mc are moments and α is the angular acceleration both about an axis normal to the plane of motion, Ic is the mass moment of inertia about the normal axis through the mass center, and ρpc is a vector from point p to point c. DYNAMICS

59

Free Vibration The figure illustrates a single degree-of-freedom system.

Torsional Vibration

Position of Undeformed Length of Spring

m

δst

kt

Position of Static Equilibrium

x

I

k θ

The equation of motion may be expressed as mxp = mg - k _ x + dst i where m is mass of the system, k is the spring constant of the system, δst is the static deflection of the system, and x is the displacement of the system from static equilibrium. From statics it may be shown that mg = kδst thus the equation of motion may be written as mxp + kx = 0, or xp + ^ k/mh x = 0

The solution of this differential equation is x(t) = C1 cos(ωnt) + C2 sin(ωnt) where ωn = k m is the undamped natural circular frequency and C1 and C2 are constants of integration whose values are determined from the initial conditions. If the initial conditions are denoted as x(0) = x0 and xo ^0h = v0, then

x ^t h = x0 cos _~nt i + _ v0 /~ni sin _~nt i

It may also be shown that the undamped natural frequency may be expressed in terms of the static deflection of the system as ~n =

g/dst

The undamped natural period of vibration may now be written as xn = 2r/~n = 2r m k = 2r dst g

60

DYNAMICS

For torsional free vibrations it may be shown that the differential equation of motion is ip + _ k I i i = 0, where t

θ = the angular displacement of the system kt = the torsional stiffness of the massless rod I = the mass moment of inertia of the end mass The solution may now be written in terms of the initial conditions i ^0h = i0 and io ^0h = io 0 as i ^t h = i cos _~ t i + _io /~ i sin _~ t i 0

n

0

n

n

where the undamped natural circular frequency is given by ~n = kt /I The torsional stiffness of a solid round rod with associated polar moment-of-inertia J, length L, and shear modulus of elasticity G is given by kt = GJ/L Thus the undamped circular natural frequency for a system with a solid round supporting rod may be written as ~n = GJ/IL Similar to the linear vibration problem, the undamped natural period may be written as xn = 2r/~n = 2r I/kt = 2r IL /GJ See second-order control-system models in the MEASUREMENT AND CONTROLS section for additional information on second-order systems.

DYNAMICS

61

z

R2

z

R1

z

z

y

R

R

y

c

y

y

c

c

cR

L

c

h

x

h

x

x

x

x

= = = = =

xc yc zc ρ = = = =

M =

4 3 πR ρ 3 0 0 0 mass/vol.

(

πR2ρh 0 h/2 0 mass/vol.

)

M = π R12 − R22 ρ h xc = 0 yc = h 2 zc = 0 ρ = mass vol.

M xc yc zc ρ

= = = = =

2πRρA R = mean radius R = mean radius 0 cross-sectional area of ring ρ = mass/vol.

M xc yc zc A

= ρLA = L/2 =0 =0 = cross-sectional area of rod ρ = mass/vol.

M xc yc zc A

Mass & Centroid

(

)

( (

)

2

)

)

) 12

= 3R

= 2

(

= 3R 2

2

)

=

rz2

2

) 12

2 1

)

+ R22 2

)

ry2c = ry2 = 2 R 2 5 rz2c =r 2z = 2 R 2 5

I zc = I z = 2 MR 2 5

rx2c = rx2 = 2 R 2 5

= 3R12 + 3R22 + 4h 2 12

(

rx2 = rz2

ry2c = ry2

( = (R

(

= 3R + 4h

2

)

rx2c = rz2c = 3R12 + 3R22 + h 2 12

rx2

ry2c = ry2 = R 2 2

rx2c = rz2c = 3R 2 + h 2 12

rz2

ry2

I yc = I y = 2 MR 2 5

I xc = I x = 2 MR 2 5

(

= M 3R12 + 3R22 + h 2 12 I y c = I y = M R12 + R22 2 Ix = Iz = M 3R12 + 3R22 + 4h 2 12

I xc = I z c

(

I x = I z = M 3R + 4h

2

I yc = I y = MR 2 2

I xc = I zc = M 3R 2 + h 2 12

I z = 3MR

2

I x = I y = 3MR 2

rx2

r22c = R 2

I zc = MR 2 2

rx2c = ry2c = R 2 2

=L 3

I xc = I yc = MR 2 2

=

2

ry2

I y = I z = ML2 3

rz2

ry2c = rz2c = L2 12

rx2 = rx2c = 0


I yc = I zc = ML2 12

I x = I xc = 0

Mass Moment of Inertia


z

y

Figure

I xc yc, etc. = 0

I xy , etc. = 0

I xc yc , etc. = 0

I xy , etc. = 0

I xc yc , etc. = 0

I xz = I yz = 0

I zc zc = MR 2

I xc yc , etc. = 0

I xy , etc. = 0

I xc yc , etc. = 0

Product of Inertia

FLUID MECHANICS DENSITY, SPECIFIC VOLUME, SPECIFIC WEIGHT, AND SPECIFIC GRAVITY The definitions of density, specific volume, specific weight, and specific gravity follow: t = limit

Dm DV

c = limit

DW DV

c = limit

g : D m D V = tg

DV " 0

DV " 0 DV " 0

also SG = γ/γw = ρ/ρw , where = density (also called mass density), t Δm = mass of infinitesimal volume, ΔV = volume of infinitesimal object considered, γ = specific weight, = ρg, ΔW = weight of an infinitesimal volume, SG = specific gravity, = density of water at standard conditions tw = 1,000 kg/m3 (62.43 lbm/ft3), and γω = specific weight of water at standard conditions, = 9,810 N/m3 (62.4 lbf/ft3), and = 9,810 kg/(m2•s2).

STRESS, PRESSURE, AND VISCOSITY Stress is defined as x ]1g = limit

DA " 0

DF/DA, where

x ]1g = surface stress vector at point 1, ΔF = force acting on infinitesimal area ΔA, and ΔA = infinitesimal area at point 1. τn = – P τt = μ(dv/dy) (one-dimensional; i.e., y), where τn and τt = the normal and tangential stress components at point 1, P = the pressure at point 1, μ = absolute dynamic viscosity of the fluid N•s/m2 [lbm/(ft-sec)], dv = differential velocity, dy = differential distance, normal to boundary. v = velocity at boundary condition, and y = normal distance, measured from boundary. v = kinematic viscosity; m2/s (ft2/sec) where v = μ/ρ

For a thin Newtonian fluid film and a linear velocity profile, v(y) = vy/δ; dv/dy = v/δ, where v = velocity of plate on film and δ = thickness of fluid film. For a power law (non-Newtonian) fluid τt = K (dv/dy)n, where K = consistency index, and n = power law index. n < 1 ≡ pseudo plastic n > 1 ≡ dilatant

SURFACE TENSION AND CAPILLARITY Surface tension σ is the force per unit contact length σ = F/L, where σ = surface tension, force/length, F = surface force at the interface, and L = length of interface. The capillary rise h is approximated by h = 4σ cos β/(γd), where h = the height of the liquid in the vertical tube, σ = the surface tension, β = the angle made by the liquid with the wetted tube wall, γ = specific weight of the liquid, and d = the diameter of the capillary tube.

THE PRESSURE FIELD IN A STATIC LIQUID ♦

The difference in pressure between two different points is P2 – P1 = –γ (z2 – z1) = –γh = –ρgh For a simple manometer, Po = P2 + γ2z2 – γ1z1 Absolute pressure = atmospheric pressure + gage pressure reading Absolute pressure = atmospheric pressure – vacuum gage pressure reading ♦ Bober, W. & R.A. Kenyon, Fluid Mechanics, Wiley, New York, 1980. Diagrams reprinted by permission of William Bober & Richard A. Kenyon.

62

FLUID MECHANICS

FORCES ON SUBMERGED SURFACES AND THE CENTER OF PRESSURE ♦ p

The pressure on a point at a distance Z′ below the surface is p = po + γZ′, for Z′ ≥ 0 If the tank were open to the atmosphere, the effects of po could be ignored. The coordinates of the center of pressure (CP) are y* = (γI yc zc sinα)/(pc A) and z* = (γI ycsinα )/(pc A), where y* =

the y-distance from the centroid (C) of area (A) to the center of pressure,

z* =

the z-distance from the centroid (C) of area (A) to the center of pressure,

I yc and I yc zc = the moment and product of inertia of the area, pc =

the pressure at the centroid of area (A), and

Zc =

the slant distance from the water surface to the centroid (C) of area (A).

♦ p

The force on a rectangular plate can be computed as F = [p1Av + (p2 – p1) Av /2]i + Vf γ f j, where F

=

force on the plate,

p1 =

pressure at the top edge of the plate area,

p2 =

pressure at the bottom edge of the plate area,

Av =

vertical projection of the plate area,

Vf =

volume of column of fluid above plate, and

γf

specific weight of the fluid.

=

ARCHIMEDES PRINCIPLE AND BUOYANCY 1. The buoyant force exerted on a submerged or floating body is equal to the weight of the fluid displaced by the body. 2. A floating body displaces a weight of fluid equal to its own weight; i.e., a floating body is in equilibrium. The center of buoyancy is located at the centroid of the displaced fluid volume. In the case of a body lying at the interface of two immiscible fluids, the buoyant force equals the sum of the weights of the fluids displaced by the body.

ONE-DIMENSIONAL FLOWS The Continuity Equation So long as the flow Q is continuous, the continuity equation, as applied to one-dimensional flows, states that the flow passing two points (1 and 2) in a stream is equal at each point, A1v1 = A2v2. Q = Av mo = ρQ = ρAv, where Q =

volumetric flow rate,

mo =

mass flow rate,

A

=

cross section of area of flow,

v

=

average flow velocity, and

ρ

=

the fluid density.

For steady, one-dimensional flow, mo is a constant. If, in addition, the density is constant, then Q is constant.

If the free surface is open to the atmosphere, then po = 0 and pc = γZc sinα.

♦ Bober, W. & R.A. Kenyon, Fluid Mechanics, Wiley, New York, 1980. Diagrams reprinted by permission of William Bober & Richard A. Kenyon.

y* = Iyc zc /(AZc) and z* = Iyc /(AZc)

FLUID MECHANICS

63

The Field Equation is derived when the energy equation is applied to one-dimensional flows. Assuming no friction losses and that no pump or turbine exists between sections 1 and 2 in the system, P2 P1 v 22 v12 z z or + + = + 2 c c 2g 2g + 1 P2 v 22 P1 v12 z z g, where g + + = + 2 t t 2 2 + 1 P1, P2 = pressure at sections 1 and 2, v1, v2 = average velocity of the fluid at the sections, z1, z2 = the vertical distance from a datum to the sections (the potential energy), γ = the specific weight of the fluid (ρg), and g = the acceleration of gravity.

FLUID FLOW The velocity distribution for laminar flow in circular tubes or between planes is v ]r g = vmax =1 - b r l G, where R 2

r R

= =

v = vmax = vmax = vmax = vmax = v =

the distance (m) from the centerline, the radius (m) of the tube or half the distance between the parallel planes, the local velocity (m/s) at r, and the velocity (m/s) at the centerline of the duct. 1.18v, for fully turbulent flow 2v, for circular tubes in laminar flow and 1.5v, for parallel planes in laminar flow, where the average velocity (m/s) in the duct.

The shear stress distribution is x r xw = R , where τ and τw are the shear stresses at radii r and R respectively. The drag force FD on objects immersed in a large body of flowing fluid or objects moving through a stagnant fluid is FD = CD = v = A

64

=

CD tv 2A , where 2

the drag coefficient, the velocity (m/s) of the flowing fluid or moving object, and the projected area (m2) of blunt objects such as spheres, ellipsoids, disks, and plates, cylinders, ellipses, and air foils with axes perpendicular to the flow.

FLUID MECHANICS

For flat plates placed parallel with the flow CD = 1.33/Re0.5 (104 < Re < 5 × 105) CD = 0.031/Re1/7 (106 < Re < 109) The characteristic length in the Reynolds Number (Re) is the length of the plate parallel with the flow. For blunt objects, the characteristic length is the largest linear dimension (diameter of cylinder, sphere, disk, etc.) which is perpendicular to the flow.

AERODYNAMICS Airfoil Theory The lift force on an airfoil is given by FL =

CL tv 2AP 2

CL = the lift coefficient v = velocity (m/s) of the undisturbed fluid and AP = the projected area of the airfoil as seen from above (plan area). This same area is used in defining the drag coefficient for an airfoil. The lift coefficient can be approximated by the equation 2πk1sin(α + β) which is valid for small values of α and β. k1 = a constant of proportionality α = angle of attack (angle between chord of airfoil and direction of flow) β = negative of angle of attack for zero lift. CL =

The drag coefficient may be approximated by C2 CD = CD3 + L rAR CD∞ = infinite span drag coefficient 2 Ap AR = b = 2 Ap c

The aerodynamic moment is given by CM tv 2Apc M= 2 where the moment is taken about the front quarter point of the airfoil. CM = moment coefficient Ap = plan area c = chord length b = span length AERODYNAMIC MOMENT CENTER CAMBER LINE

α V c 4

CHORD c

REYNOLDS NUMBER Re = vDt/n = vD/y Rel =

^ 2 - nh

The Darcy-Weisbach equation is

D t v , where n ^ n - 1h n 3 1 + l8 Kb 4n

ρ = the mass density, D = the diameter of the pipe, dimension of the fluid streamline, or characteristic length. μ = the dynamic viscosity, y = the kinematic viscosity, Re = the Reynolds number (Newtonian fluid), Re′ = the Reynolds number (Power law fluid), and K and n are defined in the Stress, Pressure, and Viscosity section. The critical Reynolds number (Re)c is defined to be the minimum Reynolds number at which a flow will turn turbulent. Flow through a pipe is generally characterized as laminar for Re < 2,100 and fully turbulent for Re > 10,000, and transitional flow for 2,100 < Re < 10,000.

HYDRAULIC GRADIENT (GRADE LINE) The hydraulic gradient (grade line) is defined as an imaginary line above a pipe so that the vertical distance from the pipe axis to the line represents the pressure head at that point. If a row of piezometers were placed at intervals along the pipe, the grade line would join the water levels in the piezometer water columns. ENERGY LINE (BERNOULLI EQUATION) The Bernoulli equation states that the sum of the pressure, velocity, and elevation heads is constant. The energy line is this sum or the “total head line” above a horizontal datum. The difference between the hydraulic grade line and the energy line is the v2/2g term. STEADY, INCOMPRESSIBLE FLOW IN CONDUITS AND PIPES The energy equation for incompressible flow is v12

v 22

p1 p2 c + z1 + 2g = c + z2 + 2g + h f or p1 p2 v12 v 22 z z h + + = + + 1 2 tg tg 2g 2g + f hf

=

2 h f = f L v , where D 2g

n

the head loss, considered a friction effect, and all remaining terms are defined above.

If the cross-sectional area and the elevation of the pipe are the same at both sections (1 and 2), then z1 = z2 and v1 = v2.

f D L e

= = = =

f(Re, e/D), the Moody or Darcy friction factor, diameter of the pipe, length over which the pressure drop occurs, roughness factor for the pipe, and all other symbols are defined as before. An alternative formulation employed by chemical engineers is 2 2f FanningLv 2 h f = ` 4f Fanningj Lv = Dg D 2g f Fanning friction factor, fFanning = 4 A chart that gives f versus Re for various values of e/D, known as a Moody or Stanton diagram, is available at the end of this section. Friction Factor for Laminar Flow The equation for Q in terms of the pressure drop Δpf is the Hagen-Poiseuille equation. This relation is valid only for flow in the laminar region. Q=

rR 4 D p f rD 4 D p f = 8nL 128nL

Flow in Noncircular Conduits Analysis of flow in conduits having a noncircular cross section uses the hydraulic diameter DH, or the hydraulic radius RH, as follows D RH = cross -sectional area = H 4 wetted perimeter Minor Losses in Pipe Fittings, Contractions, and Expansions Head losses also occur as the fluid flows through pipe fittings (i.e., elbows, valves, couplings, etc.) and sudden pipe contractions and expansions. v12 v 22 p1 p2 z z h h + + = + + 1 2 c c 2g 2g + f + f, fitting v12 v 22 p1 p2 z , where z h h + + = + + 1 2 tg tg 2g 2g + f + f, fitting 2 2 h f, fitting = C v , and v = 1 velocity head 2g 2g Specific fittings have characteristic values of C, which will be provided in the problem statement. A generally accepted nominal value for head loss in well-streamlined gradual contractions is hf, fitting = 0.04 v2/ 2g

The pressure drop p1 – p2 is given by the following: p1 – p2 = γ hf = ρghf

FLUID MECHANICS

65

The head loss at either an entrance or exit of a pipe from or to a reservoir is also given by the hf, fitting equation. Values for C for various cases are shown as follows. ♦

Jet Propulsion •

PUMP POWER EQUATION Wo = Qch/h = Qtgh/h, where Q h η Wo

= = = =

p = the internal pressure in the pipe line, A = the cross-sectional area of the pipe line, W= the weight of the fluid, v = the velocity of the fluid flow, α = the angle the pipe bend makes with the horizontal, ρ = the density of the fluid, and Q = the quantity of fluid flow.

v

volumetric flow (m3/s or cfs), head (m or ft) the fluid has to be lifted, efficiency, and power (watts or ft-lbf/sec).

v

v

For additional information on pumps refer to the MECHANICAL ENGINEERING section of this handbook.

COMPRESSIBLE FLOW See the MECHANICAL ENGINEERING section for compressible flow and machinery associated with compressible flow (compressors, turbines, fans).

F γ h A2 Q v2

THE IMPULSE-MOMENTUM PRINCIPLE The resultant force in a given direction acting on the fluid equals the rate of change of momentum of the fluid. ΣF = Q2ρ2v2 – Q1ρ1v1, where ΣF

F = Qρ(v2 – 0) = = = = = =

F = 2γhA2, where the propulsive force, the specific weight of the fluid, the height of the fluid above the outlet, the area of the nozzle tip, A2 2gh , and 2gh .

Deflectors and Blades Fixed Blade v •

= the resultant of all external forces acting on the control volume,

Q1ρ1v1 = the rate of momentum of the fluid flow entering the control volume in the same direction of the force, and Q2ρ2v2 = the rate of momentum of the fluid flow leaving the control volume in the same direction of the force. Pipe Bends, Enlargements, and Contractions The force exerted by a flowing fluid on a bend, enlargement, or contraction in a pipe line may be computed using the impulse-momentum principle. •

v

v

v

v

v

– Fx = Qρ(v2cos α – v1) Fy = Qρ(v2sin α – 0)

Moving Blade • v v

v

v

v v

v

v

v v

v

v

v v

v v

v v

v

v

v

p1A1 – p2A2cos α – Fx = Qρ (v2cos α – v1) Fy – W – p2A2sin α = Qρ (v2sin α – 0), where F = the force exerted by the bend on the fluid (the force exerted by the fluid on the bend is equal in magnitude and opposite in sign), Fx and Fy are the x-component and y-component of the force,

– Fx = Qρ(v2x – v1x) = – Qρ(v1 – v)(1 – cos α) Fy = Qρ(v2y – v1y) = + Qρ(v1 – v) sin α, where v = the velocity of the blade. ♦ Bober, W. & R.A. Kenyon, Fluid Mechanics, Wiley, New York, 1980. Diagram reprinted by permission of William Bober & Richard A. Kenyon. • Vennard, J.K., Elementary Fluid Mechanics, 6th ed., J.K. Vennard, 1954.

66

FLUID MECHANICS

Impulse Turbine • v

⋅ W ⋅ W v v

v

v

α

v

v

WEIR FORMULAS See the CIVIL ENGINEERING section.

v

v

Hazen-Williams Equation v = k1CR0.63S0.54, where C = roughness coefficient, k1 = 0.849 for SI units, and k1 = 1.318 for USCS units. Other terms defined as above.

v

Wo = Qt ^v1 - vh ^1 - cos ah v, where

FLOW THROUGH A PACKED BED

Wo max = Qt ` v12/4j ^1 - cos ah

L

Dp =

average particle diameter (m)

Wo max = `Qtv12j /2 = `Qcv12j /2g

Φs =

sphericity of particles, dimensionless (0–1)

ε

porosity or void fraction of the particle bed, dimensionless (0–1)

A porous, fixed bed of solid particles can be characterized by

Wo = power of the turbine. When a = 180c,

MULTIPATH PIPELINE PROBLEMS •

=

=

The Ergun equation can be used to estimate pressure loss through a packed bed under laminar and turbulent flow conditions.

p

2 1.75t v o ^1 - fh Dp 150vo n ^1 - fh = + 2 2 3 L UsDp f3 Us D p f 2

p

L v v

v

L

The same head loss occurs in each branch as in the combination of the two. The following equations may be solved simultaneously for vA and vB: hL = fA

length of particle bed (m)

L A v A2 L v2 fB B B = DA 2g DB 2g

_rD 2/4i v = `rD A2 /4j v A + `rDB2 /4j vB

The flow Q can be divided into QA and QB when the pipe characteristics are known.

OPEN-CHANNEL FLOW AND/OR PIPE FLOW Manning’s Equation v = (k/n)R2/3S1/2, where k = 1 for SI units, k = 1.486 for USCS units, v = velocity (m/s, ft/sec), n = roughness coefficient, R = hydraulic radius (m, ft), and S = slope of energy grade line (m/m, ft/ft). Also see Hydraulic Elements Graph for Circular Sewers in the CIVIL ENGINEERING section.

Δp =

pressure loss across packed bed (Pa)

vo =

superficial (flow through empty vessel) l fluid velocity b m s

ρ

=

fluid density d

kg n m3

μ

=

kg fluid viscosity c m : s m

FLUID MEASUREMENTS The Pitot Tube – From the stagnation pressure equation for an incompressible fluid, v = _ 2 ti _ p0 - psi = 2g _ p0 - psi /c, where v = p0 = ps =

the velocity of the fluid, the stagnation pressure, and the static pressure of the fluid at the elevation where the measurement is taken.

• V2 2g

ps

V, ps

po

For a compressible fluid, use the above incompressible fluid equation if the Mach number ≤ 0.3. •

Vennard, J.K., Elementary Fluid Mechanics, 6th ed., J.K. Vennard, 1954.

FLUID MECHANICS

67

Manometers ♦ p

Orifices The cross-sectional area at the vena contracta A2 is characterized by a coefficient of contraction Cc and given by Cc A. • 0

p

p

p p Q = CA0 2g d c1 + z1 - c2 - z2 n For a simple manometer, p0 = p2 + γ2h2 – γ1h1 = p2 + g (ρ2 h2– ρ1 h1) If h1 = h2 = h p0 = p2 + (γ2 – γ1)h = p2 + (ρ2 – ρ1)gh

where C, the coefficient of the meter (orifice coefficient), is given by C=

Note that the difference between the two densities is used. Another device that works on the same principle as the manometer is the simple barometer. patm = pA = pv + γh = pB + γh = pB + ρgh

CvCc

1 - Cc2 _ A0 A1i

2

♦

♦

pv = vapor pressure of the barometer fluid Venturi Meters Q=

CvA2

1 - ^ A2/A1h

2

p p 2g d c1 + z1 - c2 - z2 n , where

For incompressible flow through a horizontal orifice meter installation 2 _p - p i Q = CA0 t 1 2 Submerged Orifice operating under steady-flow conditions: •

Cv = the coefficient of velocity, and γ = ρg. The above equation is for incompressible fluids. •

A1

{

}A

2

Q = A2v2 = CcCv A 2g ^h1 - h2h = CA 2g ^h1 - h2h

in which the product of Cc and Cv is defined as the coefficient of discharge of the orifice. ♦ Bober, W. & R.A. Kenyon, Fluid Mechanics, Wiley, New York, 1980. Diagram reprinted by permission of William Bober & Richard A. Kenyon. • Vennard, J.K., Elementary Fluid Mechanics, 6th ed., J.K. Vennard, 1954.

68

FLUID MECHANICS

Orifice Discharging Freely into Atmosphere • Atm

SIMILITUDE In order to use a model to simulate the conditions of the prototype, the model must be geometrically, kinematically, and dynamically similar to the prototype system. To obtain dynamic similarity between two flow pictures, all independent force ratios that can be written must be the same in both the model and the prototype. Thus, dynamic similarity between two flow pictures (when all possible forces are acting) is expressed in the five simultaneous equations below.

0

Q = CA0 2gh in which h is measured from the liquid surface to the centroid of the orifice opening.

DIMENSIONAL HOMOGENEITY AND DIMENSIONAL ANALYSIS Equations that are in a form that do not depend on the fundamental units of measurement are called dimensionally homogeneous equations. A special form of the dimensionally homogeneous equation is one that involves only dimensionless groups of terms. Buckingham’s Theorem: The number of independent dimensionless groups that may be employed to describe a phenomenon known to involve n variables is equal to the number (n – r ), where r is the number of basic dimensions (i.e., M, L, T) needed to express the variables dimensionally.

•

Vennard, J.K., Elementary Fluid Mechanics, 6th ed., J.K. Vennard, 1954.

[ [ [ [ [

FI Fp FI FV FI FG FI FE FI FT

] ] ] ] ]

= p

= p

= p

= p

= p

[ [ [ [ [

FI Fp FI FV FI FG FI FE FI FT

] ] ] ] ]

= m

= m

[ [ [ [ [

= m

= m

m

=

] [ ] ] [ ] [] [] ] [ ] [] [] ] [ ] [] [] ] [ ] [ ] [ ]

ρv 2 p

vl ρ μ v2 lg

p

vl ρ μ

=

p

=

p

ρv 2 Ev

ρ lv σ

ρv 2 p

=

v2 lg

=

p

2

=

p

m

= Re p = Re

m

m

= Fr p = Fr

m

m

ρv 2 Ev

ρ lv 2 σ

= Ca

p

= Ca

m

m

= We p = We

m

m

where the subscripts p and m stand for prototype and model respectively, and FI FP FV FG FE FT Re We Ca Fr l v ρ σ Ev μ p g

= = = = = = = = = = = = = = = = = =

inertia force, pressure force, viscous force, gravity force, elastic force, surface tension force, Reynolds number, Weber number, Cauchy number, Froude number, characteristic length, velocity, density, surface tension, bulk modulus, dynamic viscosity, pressure, and acceleration of gravity.

FLUID MECHANICS

69

f

0 5 10 15 20 25 30 40 50 60 70 80 90 100

9.805 9.807 9.804 9.798 9.789 9.777 9.764 9.730 9.689 9.642 9.589 9.530 9.466 9.399

999.8 1000.0 999.7 999.1 998.2 997.0 995.7 992.2 988.0 983.2 977.8 971.8 965.3 958.4

0.001781 0.001518 0.001307 0.001139 0.001002 0.000890 0.000798 0.000653 0.000547 0.000466 0.000404 0.000354 0.000315 0.000282

0.000001785 0.000001518 0.000001306 0.000001139 0.000001003 0.000000893 0.000000800 0.000000658 0.000000553 0.000000474 0.000000413 0.000000364 0.000000326 0.000000294

0.61 0.87 1.23 1.70 2.34 3.17 4.24 7.38 12.33 19.92 31.16 47.34 70.10 101.33

PROPERTIES OF WATER (ENGLISH UNITS) Temperature (°F)

Specific Weight γ (lb/ft3)

32

6 2 . 42

Mass Density ρ (lb • sec 2/ft4 ) 1.940

Absolute Dynamic Viscosity μ (× 10 –5 lb • sec/ft 2 )

Kinematic Viscosity

υ

(× 10 –5 ft2 /sec)

Vapor Pressure pv (psi)

3 . 74 6

1.931

0.09

40

62.43

1. 940

3 . 22 9

1.664

0 . 12

50

6 2 . 41

1.940

2 . 73 5

1.410

0.18

60

6 2 . 37

1.938

2 . 35 9

1.217

0. 2 6

70

6 2 . 30

1.936

2 . 05 0

1.059

0.36

80

6 2 . 22

1.934

1 . 79 9

0.930

0.51

90

6 2 . 11

1.931

1 . 59 5

0.826

0.70

100

62.00

1. 9 2 7

1 . 4 24

0.739

0.95

110

61.86

1. 9 2 3

1 . 2 84

0.667

1.24

120

61.71

1. 9 1 8

1 . 1 68

0.609

1.69

130

61.55

1. 9 1 3

1 . 0 69

0.558

2.22

140

61.38

1. 9 0 8

0 . 9 81

0.514

2.89

150

61.20

1. 9 0 2

0 . 9 05

0.476

3.72

160

61.00

1. 8 9 6

0 . 8 38

0.442

4.74

170

60.80

1. 8 9 0

0 . 7 80

0.413

5.99

180

60.58

1. 8 8 3

0. 7 26

0.385

7.51

190

60.36

1. 8 7 6

0 . 6 78

0.362

9.34

200

60.12

1. 8 6 8

0 . 6 37

0.341

11.52

212

59.83

1. 8 6 0

0 . 5 93

0.319

14.70

a

From "Hydraulic Models,"ASCE Manual of Engineering Practice, No. 25, ASCE, 1942. From J.H. Keenan and F.G. Keyes, Thermodynamic Properties of Steam, John Wiley & Sons, 1936. f Compiled from many sources including those indicated: Handbook of Chemistry and Physics, 54th ed., The CRC Press, 1973, and Handbook of Tables for Applied Engineering Science, The Chemical Rubber Co., 1970. Vennard, J.K. and Robert L. Street, Elementary Fluid Mechanics, 6th ed., Wiley, New York, 1982. e

70

Vapor Pressuree , p v , kPa

Kinematic Viscosity a, υ m 2 /s

Absolute Dynamic a Viscosity , μ Pa•s

Density a , ρ, kg/m 3

Specific Weight a, γ, kN/m 3

Temperature °C

PROPERTIES OF WATER (SI METRIC UNITS)

FLUID MECHANICS

MOODY (STANTON) DIAGRAM

Material Riveted steel Concrete Cast iron Galvanized iron Commercial steel or wrought iron Drawn tubing

e (ft) 0.003–0.03 0.001–0.01 0.00085 0.0005 0.00015 0.000005

e (mm) 0.9–9.0 0.3–3.0 0.25 0.15 0.046 0.0015

MOODY (STANTON) FRICTION FACTOR, f

=

REYNOLDS NUMBER, Re = Dvρ μ

From ASHRAE (The American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc.)

FLUID MECHANICS

71

FLUID MECHANICS

2FD ρv 2A

CD =

24 , Re < 10 Re

Dv

DRAG COEFFICIENTS FOR SPHERES, DISKS, AND CYLINDERS

Note: Intermediate divisions are 2, 4, 6, and 8.

CD

72

THERMODYNAMICS PROPERTIES OF SINGLE-COMPONENT SYSTEMS Nomenclature 1. Intensive properties are independent of mass. 2. Extensive properties are proportional to mass. 3. Specific properties are lowercase (extensive/mass). State Functions (properties) Absolute Pressure, P Absolute Temperature, T Volume, V Specific Volume, v = V m Internal Energy, U Specific Internal Energy, u =U m Enthalpy, H Specific Enthalpy, h = u + Pv = H/m Entropy, S Specific Entropy, s = S/m Gibbs Free Energy, g = h – Ts Helmholz Free Energy, a = u – Ts

(lbf/in2 or Pa) (°R or K) (ft3 or m3) (ft3/lbm or m3/kg) (Btu or kJ) (usually in Btu/lbm or kJ/kg) (Btu or KJ) (usually in Btu/lbm or kJ/kg) (Btu/°R or kJ/K) [Btu/(lbm-°R) or kJ/(kg•K)] (usually in Btu/lbm or kJ/kg)

(usually in Btu/lbm or kJ/kg) Heat Capacity at Constant Pressure, cp = b 2h l 2T P 2 Heat Capacity at Constant Volume, cv = b u l 2T v Quality x (applies to liquid-vapor systems at saturation) is defined as the mass fraction of the vapor phase: x = mg /(mg + mf), where

mg = mass of vapor, and mf = mass of liquid. Specific volume of a two-phase system can be written: v = xvg + (1 – x)vf or v = vf + xvfg, where vf = specific volume of saturated liquid, vg = specific volume of saturated vapor, and vfg = specific volume change upon vaporization. = vg – vf Similar expressions exist for u, h, and s: u = xug + (1 – x) uf or u = uf + xufg h = xhg + (1 – x) hf or h = hf + xhfg s = xsg + (1 – x) sf or s = sf + xsfg

For an ideal gas, Pv = RT or PV = mRT, and P1v1/T1 = P2v2/T2, where P = pressure, v = specific volume, m = mass of gas, R = gas constant, and T = absolute temperature. V = volume R is specific to each gas but can be found from R R= , where ^mol. wt h R = the universal gas constant = 1,545 ft-lbf/(lbmol-°R) = 8,314 J/(kmol⋅K). For ideal gases, cp – cv = R Also, for ideal gases: b 2h l = 0 2P T

b 2u l = 0 2v T

For cold air standard, heat capacities are assumed to be constant at their room temperature values. In that case, the following are true: Δu = cvΔT; Δh = cp ΔT Δs = cp ln (T2 /T1) – R ln (P2 /P1); and Δs = cv ln (T2 /T1) + R ln (v2 /v1). For heat capacities that are temperature dependent, the value to be used in the above equations for Δh is known as the mean heat capacity `c pj and is given by T

#T 2 cpdT cp = 1 T2 - T1 Also, for constant entropy processes: T2 P d 2n T1 = P1

k

P2 v d 1n ; P1 = v2 T2 v d 1n T1 = v2

k-1 k

k-1

, where k = cp cv

For real gases, several equations of state are available; one such equation is the van der Waals equation with constants based on the critical point:

c P + a2 m ^v - bh = RT v R 2Tc p RT , b= c where a = c 27 m f 64 Pc 8Pc where Pc and Tc are the pressure and temperature at the critical point, respectively, and v is the molar specific volume. 2

For a simple substance, specification of any two intensive, independent properties is sufficient to fix all the rest.

THERMODYNAMICS

73

FIRST LAW OF THERMODYNAMICS The First Law of Thermodynamics is a statement of conservation of energy in a thermodynamic system. The net energy crossing the system boundary is equal to the change in energy inside the system. Heat Q is energy transferred due to temperature difference and is considered positive if it is inward or added to the system. Closed Thermodynamic System No mass crosses system boundary Q – W = ΔU + ΔKE + ΔPE where ΔKE = change in kinetic energy, and ΔPE = change in potential energy. Energy can cross the boundary only in the form of heat or work. Work can be boundary work, wb, or other work forms (electrical work, etc.) l Work W b w = W m is considered positive if it is outward or work done by the system. Reversible boundary work is given by wb = ∫P dv. Special Cases of Closed Systems Constant Pressure (Charles’ Law): wb = PΔv (ideal gas) T/v = constant Constant Volume: wb = 0 (ideal gas) T/P = constant Isentropic (ideal gas): Pvk = constant w = (P2v2 – P1v1)/(1 – k) = R(T2 – T1)/(1 – k) Constant Temperature (Boyle’s Law): (ideal gas) Pv = constant wb = RTln (v2 / v1) = RTln (P1/P2) Polytropic (ideal gas): Pvn = constant w = (P2v2 – P1v1)/(1 – n) Open Thermodynamic System Mass crosses the system boundary There is flow work (Pv) done by mass entering the system. The reversible flow work is given by: wrev = – ∫v dP + Δke + Δpe First Law applies whether or not processes are reversible. FIRST LAW (energy balance) Rmo i 8 hi + Vi2/2 + gZi B - Rmo e 8 he + Ve2/2 + gZe B + Qo in - Wo net = d _ msusi /dt, where

Wo net = rate of net or shaft work transfer, ms = mass of fluid within the system,

74

THERMODYNAMICS

us = specific internal energy of system, and Qo = rate of heat transfer (neglecting kinetic and potential energy of the system).

Special Cases of Open Systems Constant Volume: wrev = – v (P2 – P1) Constant Pressure: wrev = 0 Constant Temperature: (ideal gas) Pv = constant wrev = RT ln (v2 /v1) = RT ln (P1 /P2) Isentropic (ideal gas): Pvk = constant wrev = k (P2v2 – P1v1)/(1 – k) = k R (T2 – T1)/(1 – k) wrev =

^ k - 1h /k

k RT >1 - d P2 n P1 k-1 1

H

Polytropic: Pvn = constant wrev = n (P2v2 – P1v1)/(1 – n) Steady-State Systems The system does not change state with time. This assumption is valid for steady operation of turbines, pumps, compressors, throttling valves, nozzles, and heat exchangers, including boilers and condensers.

Rmo ` hi + Vi2/2 + gZi j - Rmo e ` he + Ve2/2 + gZej + Qo in - Wo out = 0

and Rmo i = Rmo e where

mo = mass flow rate (subscripts i and e refer to inlet and exit states of system), g = acceleration of gravity, Z = elevation, V = velocity, and Wo = rate of work.

Special Cases of Steady-Flow Energy Equation Nozzles, Diffusers: Velocity terms are significant. No elevation change, no heat transfer, and no work. Single mass stream. hi + Vi2/2 = he + Ve2/2 Isentropic Efficiency (nozzle) =

Ve2 - Vi2 , where 2 _ hi - hesi

hes = enthalpy at isentropic exit state. Turbines, Pumps, Compressors: Often considered adiabatic (no heat transfer). Velocity terms usually can be ignored. There are significant work terms and a single mass stream. hi = he + w

Isentropic Efficiency (turbine) =

hi - he hi - hes

Isentropic Efficiency (compressor, pump) =

hes - hi he - hi

Throttling Valves and Throttling Processes: No work, no heat transfer, and single-mass stream. Velocity terms are often insignificant. hi = he Boilers, Condensers, Evaporators, One Side in a Heat Exchanger: Heat transfer terms are significant. For a singlemass stream, the following applies: hi + q = he Heat Exchangers: No heat or work. Two separate flow rates mo 1 and mo 2 : mo 1 _ h1i - h1ei = mo 2 _ h2e - h2i i See MECHANICAL ENGINEERING section.

IDEAL GAS MIXTURES i = 1, 2, …, n constituents. Each constituent is an ideal gas. Mole Fraction: xi = Ni /N; N = Σ Ni; Σ xi = 1 where Ni = number of moles of component i.

Mass Fraction: yi = mi /m; m = Σ mi; Σ yi = 1 Molecular Weight: M = m/N = Σ xiMi Gas Constant: R = R/M To convert mole fractions xi to mass fractions yi: yi =

xi Mi R _ xi Mi i

To convert mass fractions to mole fractions: xi =

yi Mi R _ yi Mi i

Partial Pressures: P = RPi; Pi =

Mixers, Separators, Open or Closed Feedwater Heaters: Rmo ihi = Rmo ehe Rmo i = Rmo e

and

BASIC CYCLES Heat engines take in heat QH at a high temperature TH, produce a net amount of work W, and reject heat QL at a low temperature TL. The efficiency η of a heat engine is given by: η = W/QH = (QH – QL)/QH The most efficient engine possible is the Carnot Cycle. Its efficiency is given by: ηc = (TH – TL)/TH, where TH and TL = absolute temperatures (Kelvin or Rankine). The following heat-engine cycles are plotted on P-v and T-s diagrams (see later in this chapter): Carnot, Otto, Rankine Refrigeration cycles are the reverse of heat-engine cycles. Heat is moved from low to high temperature requiring work, W. Cycles can be used either for refrigeration or as heat pumps. Coefficient of Performance (COP) is defined as: COP = QH /W for heat pumps, and as COP = QL/W for refrigerators and air conditioners. Upper limit of COP is based on reversed Carnot Cycle: COPc = TH /(TH – TL) for heat pumps and COPc = TL /(TH – TL) for refrigeration.

Partial Volumes: V = !Vi; Vi =

miRiT V

mi Ri T , where P

P, V, T = the pressure, volume, and temperature of the mixture. xi = Pi /P = Vi /V

Other Properties: u = Σ (yiui); h = Σ (yihi); s = Σ (yisi) ui and hi are evaluated at T, and si is evaluated at T and Pi. PSYCHROMETRICS We deal here with a mixture of dry air (subscript a) and water vapor (subscript v): P = Pa + Pv Specific Humidity (absolute humidity, humidity ratio) ω: ω= mv /ma, where mv = mass of water vapor and ma = mass of dry air. ω = 0.622Pv /Pa = 0.622Pv /(P – Pv) Relative Humidity (rh) φ: φ = Pv /Pg , where Pg = saturation pressure at T. Enthalpy h: h = ha + ωhv Dew-Point Temperature Tdp: Tdp = Tsat at Pg = Pv

1 ton refrigeration = 12,000 Btu/hr = 3,516 W

THERMODYNAMICS

75

Wet-bulb temperature Twb is the temperature indicated by a thermometer covered by a wick saturated with liquid water and in contact with moving air. Humid Volume: Volume of moist air/mass of dry air. Psychrometric Chart A plot of specific humidity as a function of dry-bulb temperature plotted for a value of atmospheric pressure. (See chart at end of section.)

PHASE RELATIONS Clapeyron Equation for Phase Transitions: h s b dP l = fg = v fg , where dT sat Tv fg fg

Incomplete Combustion Some carbon is burned to create carbon monoxide (CO). Air-Fuel Ratio (A/F): A/F = mass of air mass of fuel Stoichiometric (theoretical) air-fuel ratio is the air-fuel ratio calculated from the stoichiometric combustion equation. Percent Theoretical Air =

Percent Excess Air =

_ A F iactual

_ A F istoichiometric

# 100

_ A F iactual - _ A F istoichiometric _ A F istoichiometric

hfg = enthalpy change for phase transitions,

SECOND LAW OF THERMODYNAMICS Thermal Energy Reservoirs ΔSreservoir = Q/Treservoir, where

vfg = volume change,

Q is measured with respect to the reservoir.

sfg = entropy change, T = absolute temperature, and (dP/dT)sat = slope of phase transition (e.g.,vapor-liquid) saturation line. Clausius-Clapeyron Equation This equation results if it is assumed that (1) the volume change (vfg) can be replaced with the vapor volume (vg), (2) the latter can be replaced with P R T from the ideal gas law, and (3) hfg is independent of the temperature (T). h fg T2 - T1 P : lne d 2 n = P1 TT 1 2 R Gibbs Phase Rule (non-reacting systems) P + F = C + 2, where P = number of phases making up a system F = degrees of freedom, and C = number of components in a system

COMBUSTION PROCESSES First, the combustion equation should be written and balanced. For example, for the stoichiometric combustion of methane in oxygen: CH4 + 2 O2 → CO2 + 2 H2O Combustion in Air For each mole of oxygen, there will be 3.76 moles of nitrogen. For stoichiometric combustion of methane in air: CH4 + 2 O2 + 2(3.76) N2 → CO2 + 2 H2O + 7.52 N2 Combustion in Excess Air The excess oxygen appears as oxygen on the right side of the combustion equation.

76

THERMODYNAMICS

# 100

Kelvin-Planck Statement of Second Law No heat engine can operate in a cycle while transferring heat with a single heat reservoir. COROLLARY to Kelvin-Planck: No heat engine can have a higher efficiency than a Carnot Cycle operating between the same reservoirs. Clausius’ Statement of Second Law No refrigeration or heat pump cycle can operate without a net work input. COROLLARY: No refrigerator or heat pump can have a higher COP than a Carnot Cycle refrigerator or heat pump.

VAPOR-LIQUID MIXTURES Henry’s Law at Constant Temperature At equilibrium, the partial pressure of a gas is proportional to its concentration in a liquid. Henry’s Law is valid for low concentrations; i.e., x ≈ 0. Pi = Pyi = hxi, where h = Henry’s Law constant, Pi = partial pressure of a gas in contact with a liquid, xi = mol fraction of the gas in the liquid, yi = mol fraction of the gas in the vapor, and P = total pressure. Raoult’s Law for Vapor-Liquid Equilibrium Valid for concentrations near 1; i.e., xi ≈1. Pi = xi Pi*, where Pi = partial pressure of component i, xi = mol fraction of component i in the liquid, and Pi* = vapor pressure of pure component i at the temperature of the mixture.

ENTROPY ds = _1 T i dQrev

EXERGY Exergy is the portion of total energy available to do work.

Inequality of Clausius

Closed-System Exergy (Availability) (no chemical reactions) φ= (u – uo) – To (s – so) + po (v – vo) where the subscript o designates environmental conditions wreversible = φ1 – φ2

s2 - s1 = #1 _1 T i dQrev 2

# _1 T i dQrev # 0

#1 _1 T i dQ # s2 - s1 2

Open-System Exergy (Availability) ψ= (h – ho) – To (s – so) + V 2/2 + gz wreversible = ψ1 – ψ2

Isothermal, Reversible Process Δs = s2 – s1 = Q/T

Gibbs Free Energy, ΔG Energy released or absorbed in a reaction occurring reversibly at constant pressure and temperature.

Isentropic Process Δs = 0; ds = 0 A reversible adiabatic process is isentropic.

Helmholtz Free Energy, ΔA Energy released or absorbed in a reaction occurring reversibly at constant volume and temperature.

Adiabatic Process δQ = 0; Δs ≥ 0 Increase of Entropy Principle Dstotal = Dssystem + Dssurroundings $ 0

Dso total = Rmo outsout - Rmo in sin - R _Qo external /Texternali $ 0 Temperature-Entropy (T-s) Diagram T 2

2

Qrev = ∫1 T d s

1 AREA = HEAT

s Entropy Change for Solids and Liquids ds = c (dT/T) s2 – s1 = ∫c (dT/T) = cmeanln (T2 /T1), where c equals the heat capacity of the solid or liquid. Irreversibility I = wrev – wactual

THERMODYNAMICS

77

COMMON THERMODYNAMIC CYCLES Carnot Cycle

Reversed Carnot

Otto Cycle (Gasoline Engine) q=0

η = 1 – r1–k r = v 1/v2

Rankine Cycle

Refrigeration (Reversed Rankine Cycle) q

wT q

wc

q q p2 = p3

p2 = p3

η=

78

THERMODYNAMICS

(h3 − h4 ) − ( h 2 − h1) h3 − h2

COP ref =

h 1 − h4 h 2 − h1

COP HP =

h2 − h3 h 2 − h1

STEAM TABLES Saturated Water - Temperature Table Temp. o C T

Sat. Press. kPa psat

0.01 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95

0.6113 0.8721 1.2276 1.7051 2.339 3.169 4.246 5.628 7.384 9.593 12.349 15.758 19.940 25.03 31.19 38.58 47.39 57.83 70.14 84.55

Specific Volume m3/kg Sat. Sat. liquid vapor vf vg 0.001 000 0.001 000 0.001 000 0.001 001 0.001 002 0.001 003 0.001 004 0.001 006 0.001 008 0.001 010 0.001 012 0.001 015 0.001 017 0.001 020 0.001 023 0.001 026 0.001 029 0.001 033 0.001 036 0.001 040

206.14 147.12 106.38 77.93 57.79 43.36 32.89 25.22 19.52 15.26 12.03 9.568 7.671 6.197 5.042 4.131 3.407 2.828 2.361 1.982

Internal Energy kJ/kg Sat. Sat. Evap. liquid vapor ufg uf ug

Enthalpy kJ/kg Sat. liquid hf

Entropy kJ/(kg·K)

sfg

Sat. vapor sg

0.00 20.97 42.00 62.99 83.95 104.88 125.78 146.67 167.56 188.44 209.32 230.21 251.11 272.02 292.95 313.90 334.86 355.84 376.85 397.88

2375.3 2361.3 2347.2 2333.1 2319.0 2304.9 2290.8 2276.7 2262.6 2248.4 2234.2 2219.9 2205.5 2191.1 2176.6 2162.0 2147.4 2132.6 2117.7 2102.7

2375.3 2382.3 2389.2 2396.1 2402.9 2409.8 2416.6 2423.4 2430.1 2436.8 2443.5 2450.1 2456.6 2463.1 2569.6 2475.9 2482.2 2488.4 2494.5 2500.6

0.01 20.98 42.01 62.99 83.96 104.89 125.79 146.68 167.57 188.45 209.33 230.23 251.13 272.06 292.98 313.93 334.91 355.90 376.92 397.96

2501.3 2489.6 2477.7 2465.9 2454.1 2442.3 2430.5 2418.6 2406.7 2394.8 2382.7 2370.7 2358.5 2346.2 2333.8 2321.4 2308.8 2296.0 2283.2 2270.2

2501.4 2510.6 2519.8 2528.9 2538.1 2547.2 2556.3 2565.3 2574.3 2583.2 2592.1 2600.9 2609.6 2618.3 2626.8 2635.3 2643.7 2651.9 2660.1 2668.1

0.0000 0.0761 0.1510 0.2245 0.2966 0.3674 0.4369 0.5053 0.5725 0.6387 0.7038 0.7679 0.8312 0.8935 0.9549 1.0155 1.0753 1.1343 1.1925 1.2500

9.1562 8.9496 8.7498 8.5569 8.3706 8.1905 8.0164 7.8478 7.6845 7.5261 7.3725 7.2234 7.0784 6.9375 6.8004 6.6669 6.5369 6.4102 6.2866 6.1659

9.1562 9.0257 8.9008 8.7814 8.6672 8.5580 8.4533 8.3531 8.2570 8.1648 8.0763 7.9913 7.9096 7.8310 7.7553 7.6824 7.6122 7.5445 7.4791 7.4159

418.94 440.02 461.14 482.30 503.50 524.74 546.02 567.35 588.74 610.18 631.68 653.24 674.87 696.56 718.33 740.17 762.09 784.10 806.19 828.37 850.65 873.04 895.53 918.14 940.87 963.73 986.74 1009.89 1033.21 1056.71 1080.39 1104.28 1128.39 1152.74 1177.36 1202.25 1227.46 1253.00 1278.92 1305.2 1332.0 1359.3 1387.1 1415.5 1444.6 1505.3 1570.3 1641.9 1725.2 1844.0 2029.6

2087.6 2072.3 2057.0 2041.4 2025.8 2009.9 1993.9 1977.7 1961.3 1944.7 1927.9 1910.8 1893.5 1876.0 1858.1 1840.0 1821.6 1802.9 1783.8 1764.4 1744.7 1724.5 1703.9 1682.9 1661.5 1639.6 1617.2 1594.2 1570.8 1546.7 1522.0 1596.7 1470.6 1443.9 1416.3 1387.9 1358.7 1328.4 1297.1 1264.7 1231.0 1195.9 1159.4 1121.1 1080.9 993.7 894.3 776.6 626.3 384.5 0

2506.5 2512.4 2518.1 2523.7 2529.3 2534.6 2539.9 2545.0 2550.0 2554.9 2559.5 2564.1 2568.4 2572.5 2576.5 2580.2 2583.7 2587.0 2590.0 2592.8 2595.3 2597.5 2599.5 2601.1 2602.4 2603.3 2603.9 2604.1 2604.0 2603.4 2602.4 2600.9 2599.0 2596.6 2593.7 2590.2 2586.1 2581.4 2576.0 2569.9 2563.0 2555.2 2546.4 2536.6 2525.5 2498.9 2464.6 2418.4 2351.5 2228.5 2029.6

419.04 440.15 461.30 482.48 503.71 524.99 546.31 567.69 589.13 610.63 632.20 653.84 675.55 697.34 719.21 741.17 763.22 785.37 807.62 829.98 852.45 875.04 897.76 920.62 943.62 966.78 990.12 1013.62 1037.32 1061.23 1085.36 1109.73 1134.37 1159.28 1184.51 1210.07 1235.99 1262.31 1289.07 1316.3 1344.0 1372.4 1401.3 1431.0 1461.5 1525.3 1594.2 1670.6 1760.5 1890.5 2099.3

2257.0 2243.7 2230.2 2216.5 2202.6 2188.5 2174.2 2159.6 2144.7 2129.6 2114.3 2098.6 2082.6 2066.2 2049.5 2032.4 2015.0 1997.1 1978.8 1960.0 1940.7 1921.0 1900.7 1879.9 1858.5 1836.5 1813.8 1790.5 1766.5 1741.7 1716.2 1689.8 1662.5 1634.4 1605.2 1574.9 1543.6 1511.0 1477.1 1441.8 1404.9 1366.4 1326.0 1283.5 1238.6 1140.6 1027.9 893.4 720.3 441.6 0

2676.1 2683.8 2691.5 2699.0 2706.3 2713.5 2720.5 2727.3 2733.9 2740.3 2746.5 2752.4 2758.1 2763.5 2768.7 2773.6 2778.2 2782.4 2786.4 2790.0 2793.2 2796.0 2798.5 2800.5 2802.1 2803.3 2804.0 2804.2 2803.8 2803.0 2801.5 2799.5 2796.9 2793.6 2789.7 2785.0 2779.6 2773.3 2766.2 2758.1 2749.0 2738.7 2727.3 2714.5 2700.1 2665.9 2622.0 2563.9 2481.0 2332.1 2099.3

1.3069 1.3630 1.4185 1.4734 1.5276 1.5813 1.6344 1.6870 1.7391 1.7907 1.8418 1.8925 1.9427 1.9925 2.0419 2.0909 2.1396 2.1879 2.2359 2.2835 2.3309 2.3780 2.4248 2.4714 2.5178 2.5639 2.6099 2.6558 2.7015 2.7472 2.7927 2.8383 2.8838 2.9294 2.9751 3.0208 3.0668 3.1130 3.1594 3.2062 3.2534 3.3010 3.3493 3.3982 3.4480 3.5507 3.6594 3.7777 3.9147 4.1106 4.4298

6.0480 5.9328 5.8202 5.7100 5.6020 5.4962 5.3925 5.2907 5.1908 5.0926 4.9960 4.9010 4.8075 4.7153 4.6244 4.5347 4.4461 4.3586 4.2720 4.1863 4.1014 4.0172 3.9337 3.8507 3.7683 3.6863 3.6047 3.5233 3.4422 3.3612 3.2802 3.1992 3.1181 3.0368 2.9551 2.8730 2.7903 2.7070 2.6227 2.5375 2.4511 2.3633 2.2737 2.1821 2.0882 1.8909 1.6763 1.4335 1.1379 0.6865 0

7.3549 7.2958 7.2387 7.1833 7.1296 7.0775 7.0269 6.9777 6.9299 6.8833 6.8379 6.7935 6.7502 6.7078 6.6663 6.6256 6.5857 6.5465 6.5079 6.4698 6.4323 6.3952 6.3585 6.3221 6.2861 6.2503 6.2146 6.1791 6.1437 6.1083 6.0730 6.0375 6.0019 5.9662 5.9301 5.8938 5.8571 5.8199 5.7821 5.7437 5.7045 5.6643 5.6230 5.5804 5.5362 5.4417 5.3357 5.2112 5.0526 4.7971 4.4298

Evap. hfg

Sat. vapor hg

Sat. liquid sf

Evap.

MPa 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200 205 210 215 220 225 230 235 240 245 250 255 260 265 270 275 280 285 290 295 300 305 310 315 320 330 340 350 360 370 374.14

0.101 35 0.120 82 0.143 27 0.169 06 0.198 53 0.2321 0.2701 0.3130 0.3613 0.4154 0.4758 0.5431 0.6178 0.7005 0.7917 0.8920 1.0021 1.1227 1.2544 1.3978 1.5538 1.7230 1.9062 2.104 2.318 2.548 2.795 3.060 3.344 3.648 3.973 4.319 4.688 5.081 5.499 5.942 6.412 6.909 7.436 7.993 8.581 9.202 9.856 10.547 11.274 12.845 14.586 16.513 18.651 21.03 22.09

0.001 044 0.001 048 0.001 052 0.001 056 0.001 060 0.001 065 0.001 070 0.001 075 0.001 080 0.001 085 0.001 091 0.001 096 0.001 102 0.001 108 0.001 114 0.001 121 0.001 127 0.001 134 0.001 141 0.001 149 0.001 157 0.001 164 0.001 173 0.001 181 0.001 190 0.001 199 0.001 209 0.001 219 0.001 229 0.001 240 0.001 251 0.001 263 0.001 276 0.001 289 0.001 302 0.001 317 0.001 332 0.001 348 0.001 366 0.001 384 0.001 404 0.001 425 0.001 447 0.001 472 0.001 499 0.001 561 0.001 638 0.001 740 0.001 893 0.002 213 0.003 155

1.6729 1.4194 1.2102 1.0366 0.8919 0.7706 0.6685 0.5822 0.5089 0.4463 0.3928 0.3468 0.3071 0.2727 0.2428 0.2168 0.194 05 0.174 09 0.156 54 0.141 05 0.127 36 0.115 21 0.104 41 0.094 79 0.086 19 0.078 49 0.071 58 0.065 37 0.059 76 0.054 71 0.050 13 0.045 98 0.042 21 0.038 77 0.035 64 0.032 79 0.030 17 0.027 77 0.025 57 0.023 54 0.021 67 0.019 948 0.018 350 0.016 867 0.015 488 0.012 996 0.010 797 0.008 813 0.006 945 0.004 925 0.003 155

THERMODYNAMICS

79

Superheated Water Tables T Temp. o C

v m3/kg

Sat. 50 100 150 200 250 300 400 500 600 700 800 900 1000 1100 1200 1300

14.674 14.869 17.196 19.512 21.825 24.136 26.445 31.063 35.679 40.295 44.911 49.526 54.141 58.757 63.372 67.987 72.602

u h kJ/kg kJ/kg p = 0.01 MPa (45.81oC) 2437.9 2443.9 2515.5 2587.9 2661.3 2736.0 2812.1 2968.9 3132.3 3302.5 3479.6 3663.8 3855.0 4053.0 4257.5 4467.9 4683.7

2584.7 2592.6 2687.5 2783.0 2879.5 2977.3 3076.5 3279.6 3489.1 3705.4 3928.7 4159.0 4396.4 4640.6 4891.2 5147.8 5409.7

s kJ/(kg⋅K) 8.1502 8.1749 8.4479 8.6882 8.9038 9.1002 9.2813 9.6077 9.8978 10.1608 10.4028 10.6281 10.8396 11.0393 11.2287 11.4091 11.5811

v m3/kg

1.6940 1.6958 1.9364 2.172 2.406 2.639 3.103 3.565 4.028 4.490 4.952 5.414 5.875 6.337 6.799 7.260

2506.1 2506.7 2582.8 2658.1 2733.7 2810.4 2967.9 3131.6 3301.9 3479.2 3663.5 3854.8 4052.8 4257.3 4467.7 4683.5

2675.5 2676.2 2776.4 2875.3 2974.3 3074.3 3278.2 3488.1 3704.4 3928.2 4158.6 4396.1 4640.3 4891.0 5147.6 5409.5

2483.9

2645.9

7.5939

3.418 3.889 4.356 4.820 5.284 6.209 7.134 8.057 8.981 9.904 10.828 11.751 12.674 13.597 14.521

2511.6 2585.6 2659.9 2735.0 2811.3 2968.5 3132.0 3302.2 3479.4 3663.6 3854.9 4052.9 4257.4 4467.8 4683.6

2682.5 2780.1 2877.7 2976.0 3075.5 3278.9 3488.7 3705.1 3928.5 4158.9 4396.3 4640.5 4891.1 5147.7 5409.6

7.6947 7.9401 8.1580 8.3556 8.5373 8.8642 9.1546 9.4178 9.6599 9.8852 10.0967 10.2964 10.4859 10.6662 10.8382

p = 0.20 MPa (120.23oC) 7.3594 7.3614 7.6134 7.8343 8.0333 8.2158 8.5435 8.8342 9.0976 9.3398 9.5652 9.7767 9.9764 10.1659 10.3463 10.5183

0.8857

2529.5

2706.7

7.1272

0.9596 1.0803 1.1988 1.3162 1.5493 1.7814 2.013 2.244 2.475 2.705 2.937 3.168 3.399 3.630

2576.9 2654.4 2731.2 2808.6 2966.7 3130.8 3301.4 3478.8 3663.1 3854.5 4052.5 4257.0 4467.5 4683.2

2768.8 2870.5 2971.0 3071.8 3276.6 3487.1 3704.0 3927.6 4158.2 4395.8 4640.0 4890.7 5147.5 5409.3

7.2795 7.5066 7.7086 7.8926 8.2218 8.5133 8.7770 9.0194 9.2449 9.4566 9.6563 9.8458 10.0262 10.1982

p = 0.40 MPa (143.63oC) Sat. 150 200 250 300 350 400 500 600 700 800 900 1000 1100 1200 1300

0.4625 0.4708 0.5342 0.5951 0.6548 0.7137 0.7726 0.8893 1.0055 1.1215 1.2372 1.3529 1.4685 1.5840 1.6996 1.8151

Sat. 200 250 300 350 400 500 600 700 800 900 1000 1100 1200 1300

0.2404 0.2608 0.2931 0.3241 0.3544 0.3843 0.4433 0.5018 0.5601 0.6181 0.6761 0.7340 0.7919 0.8497 0.9076

2553.6 2564.5 2646.8 2726.1 2804.8 2884.6 2964.4 3129.2 3300.2 3477.9 3662.4 3853.9 4052.0 4256.5 4467.0 4682.8

2738.6 2752.8 2860.5 2964.2 3066.8 3170.1 3273.4 3484.9 3702.4 3926.5 4157.3 4395.1 4639.4 4890.2 5146.8 5408.8

p = 0.60 MPa (158.85oC) 6.8959 6.9299 7.1706 7.3789 7.5662 7.7324 7.8985 8.1913 8.4558 8.6987 8.9244 9.1362 9.3360 9.5256 9.7060 9.8780

0.3157

2567.4

2756.8

6.7600

0.3520 0.3938 0.4344 0.4742 0.5137 0.5920 0.6697 0.7472 0.8245 0.9017 0.9788 1.0559 1.1330 1.2101

2638.9 2720.9 2801.0 2881.2 2962.1 3127.6 3299.1 3477.0 3661.8 3853.4 4051.5 4256.1 4466.5 4682.3

2850.1 2957.2 3061.6 3165.7 3270.3 3482.8 3700.9 3925.3 4156.5 4394.4 4638.8 4889.6 5146.3 5408.3

6.9665 7.1816 7.3724 7.5464 7.7079 8.0021 8.2674 8.5107 8.7367 8.9486 9.1485 9.3381 9.5185 9.6906

6.6628 6.8158 7.0384 7.2328 7.4089 7.5716 7.8673 8.1333 8.3770 8.6033 8.8153 9.0153 9.2050 9.3855 9.5575

0.194 44 0.2060 0.2327 0.2579 0.2825 0.3066 0.3541 0.4011 0.4478 0.4943 0.5407 0.5871 0.6335 0.6798 0.7261

p = 0.80 MPa (170.43oC)

80

THERMODYNAMICS

2576.8 2630.6 2715.5 2797.2 2878.2 2959.7 3126.0 3297.9 3476.2 3661.1 3852.8 4051.0 4255.6 4466.1 4681.8

2769.1 2839.3 2950.0 3056.5 3161.7 3267.1 3480.6 3699.4 3924.2 4155.6 4393.7 4638.2 4889.1 5145.9 5407.9

s kJ/(kg⋅K)

3.240

p = 0.10 MPa (99.63oC) Sat. 100 150 200 250 300 400 500 600 700 800 900 1000 1100 1200 1300

u h kJ/kg kJ/kg p = 0.05 MPa (81.33oC)

p = 1.00 MPa (179.91oC) 2583.6 2621.9 2709.9 2793.2 2875.2 2957.3 3124.4 3296.8 3475.3 3660.4 3852.2 4050.5 4255.1 4465.6 4681.3

2778.1 2827.9 2942.6 3051.2 3157.7 3263.9 3478.5 3697.9 3923.1 4154.7 4392.9 4637.6 4888.6 5145.4 5407.4

6.5865 6.6940 6.9247 7.1229 7.3011 7.4651 7.7622 8.0290 8.2731 8.4996 8.7118 8.9119 9.1017 9.2822 9.4543

P-h DIAGRAM FOR REFRIGERANT HFC-134a (metric units) (Reproduced by permission of the DuPont Company)

THERMODYNAMICS

81

ASHRAE PSYCHROMETRIC CHART NO. 1 (metric units) (Reproduced by permission of ASHRAE)

82

THERMODYNAMICS

THERMAL AND PHYSICAL PROPERTY TABLES (at room temperature) GASES Substance

cp

Mol wt

kJ/(kg·K)

cv

R k

Btu/(lbm-° R)

kJ/(kg·K)

Btu/(lbm-° R)

0.240 0.125 0.415 0.203 0.249

0.718 0.312 1.57 0.657 0.744

0.171 0.0756 0.381 0.158 0.178

1.40 1.67 1.09 1.29 1.40

0.2870 0.2081 0.1430 0.1889 0.2968

0.427 1.25 3.43 0.532 0.246

1.49 3.12 10.2 1.74 0.618

0.361 0.753 2.44 0.403 0.148

1.18 1.67 1.40 1.30 1.67

0.2765 2.0769 4.1240 0.5182 0.4119

0.248 0.409 0.219 0.407 0.445

0.743 1.64 0.658 1.49 1.41

0.177 0.392 0.157 0.362 0.335

1.40 1.04 1.40 1.12 1.33

0.2968 0.0729 0.2598 0.1885 0.4615

kJ/(kg·K)

Gases Air Argon Butane Carbon dioxide Carbon monoxide

29 40 58 44 28

Ethane Helium Hydrogen Methane Neon

30 4 2 16 20

Nitrogen Octane vapor Oxygen Propane Steam

1.00 0.520 1.72 0.846 1.04 1.77 5.19 14.3 2.25 1.03

28 114 32 44 18

1.04 1.71 0.918 1.68 1.87

SELECTED LIQUIDS AND SOLIDS cp

Density

Substance kJ/(kg·K)

Btu/(lbm-° R)

kg/m3

lbm/ft3

Liquids Ammonia Mercury Water

4.80 0.139 4.18

1.146 0.033 1.000

602 13,560 997

38 847 62.4

0.900 0.386 2.11 0.450 0.128

0.215 0.092 0.502 0.107 0.030

2,700 8,900 917 7,840 11,310

170 555 57.2 490 705

Solids Aluminum Copper Ice (0°C; 32°F) Iron Lead

THERMODYNAMICS

83

HEAT TRANSFER There are three modes of heat transfer: conduction, convection, and radiation.

Conduction Through a Cylindrical Wall

Conduction Fourier’s Law of Conduction Qo =- kA dT , where dx oQ = rate of heat transfer (W) k = the thermal conductivity [W/(m•K)] A = the surface area perpendicular to direction of heat transfer (m2)

T2 r1 k r2 Cylinder (Length = L)

2rkL ^T1 - T2h Qo = r ln d r2 n 1

Convection Newton’s Law of Cooling Qo = hA _Tw - T3i, where h = the convection heat transfer coefficient of the fluid [W/(m2•K)] A = the convection surface area (m2) Tw = the wall surface temperature (K) T∞ = the bulk fluid temperature (K) Radiation The radiation emitted by a body is given by Qo = fvAT 4, where ε = the emissivity of the body σ = the Stefan-Boltzmann constant = 5.67 × 10-8 W/(m2•K4) A = the body surface area (m2) T = the absolute temperature (K)

Critical Insulation Radius k rcr = insulation h3 r insulation

k insulation

Thermal Resistance (R) Qo = DT Rtotal

Plane Wall Conduction Resistance (K/W): R = L , where kA L = wall thickness r ln d r2 n 1 Cylindrical Wall Conduction Resistance (K/W): R = , 2rkL where L = cylinder length Convection Resistance (K/W) : R = 1 hA

Conduction Through a Plane Wall

k

Composite Plane Wall

T2

Fluid 1 T∞1 T1 h1

Q L

- kA ^T2 - T1h Qo = , where L A L T1 T2

h∞

Resistances in series are added: Rtotal = RR, where

CONDUCTION

T1

Q

T1

BASIC HEAT TRANSFER RATE EQUATIONS

kA

Fluid 2 T∞2 h2

kB T2

Q

= wall surface area normal to heat flow (m2) = wall thickness (m) = temperature of one surface of the wall (K) = temperature of the other surface of the wall (K)

T3 1 h1 A

LA

LB

LA kA A

LB kB A

1 h2 A

Q T∞1

84

HEAT TRANSFER

T1

T2

T3

T∞2

To evaluate Surface or Intermediate Temperatures: T2 - T3 T - T2 Qo = 1 RA = RB

Transient Conduction Using the Lumped Capacitance Method The lumped capacitance method is valid if

Steady Conduction with Internal Energy Generation The equation for one-dimensional steady conduction is o d 2T + Qgen = 0, where 2 k dx Qo gen = the heat generation rate per unit volume (W/m3) For a Plane Wall

Biot number, Bi = hV % 1, where kAs h = the convection heat transfer coefficient of the fluid [W/(m2•K)] V = the volume of the body (m3) k = thermal conductivity of the body [W/(m•K)] As = the surface area of the body (m2) Fluid h, T∞ Body

T(x) Ts2

Ts1 k

Q gen

Q"2

Q"1 x −L

T ^ xh =

As

0

L

2 Qo gen L2 T T T T d1 - x 2 n + c s2 - s1 m b x l + c s1 - s2 m L 2 2 2k L

Qo 1" + Qo 2" = 2Qo gen L, where Qo " = the rate of heat transfer per area (heat flux) (W/m2) Qo 1" = k b dT l and Qo 2" = k b dT l dx - L dx L For a Long Circular Cylinder

Constant Fluid Temperature If the temperature may be considered uniform within the body at any time, the heat transfer rate at the body surface is given by Qo = hAs ^T - T3h =- tV ^cP h b dT l, where dt T T∞ ρ cP t

= the body temperature (K) = the fluid temperature (K) = the density of the body (kg/m3) = the heat capacity of the body [J/(kg•K)] = time (s)

The temperature variation of the body with time is T - T3 = _Ti - T3i e-bt, where hAs b= tVcP

Ts

ρ, V, c P, T

where b = 1x and x = time constant ^ sh

The total heat transferred (Qtotal) up to time t is Qtotal = tVcP _Ti - T i, where

Q gen

Ti = initial body temperature (K)

r0 Q'

o 1 d b r dT l + Qgen = 0 r dr dr k T ]r g =

2 Qo genr02 1 - r 2 p + Ts f 4k r0

Qo l = rr02Qo gen, where Qo l = the heat transfer rate from the cylinder per unit length of the cylinder (W/m) HEAT TRANSFER

85

Variable Fluid Temperature If the ambient fluid temperature varies periodically according to the equation T3 = T3, mean + 1 _T3, max - T3, min i cos ^~t h 2 The temperature of the body, after initial transients have died away, is b ; 1 _T3, max - T3, min iE T= 2 cos =~t - tan- 1 c ~ mG + T3, mean b ~2 + b2 Fins For a straight fin with uniform cross section (assuming negligible heat transfer from tip), Qo = hPkAc _Tb - T3i tanh _mLci, where h P k Ac Tb T∞

= the convection heat transfer coefficient of the fluid [W/(m2•K)] = perimeter of exposed fin cross section (m) = fin thermal conductivity [W/(m•K)] = fin cross-sectional area (m2) = temperature at base of fin (K) = fluid temperature (K) hP kAc

m=

CONVECTION Terms D = diameter (m) h = average convection heat transfer coefficient of the fluid [W/(m2•K)] L = length (m) Nu = average Nusselt number c n Pr = Prandtl number = P k um = mean velocity of fluid (m/s) u∞ = free stream velocity of fluid (m/s) μ = dynamic viscosity of fluid [kg/(s•m)] ρ = density of fluid (kg/m3) External Flow In all cases, evaluate fluid properties at average temperature between that of the body and that of the flowing fluid. Flat Plate of Length L in Parallel Flow ReL =

tu3 L n

NuL = hL = 0.6640 Re1L 2 Pr1 k

3

NuL = hL = 0.0366 Re0L.8 Pr1 k

3

_ReL < 105i

_ReL > 105i

Cylinder of Diameter D in Cross Flow

A Lc = L + c , corrected length of fin (m) P

ReD =

Rectangular Fin

tu3 D n

NuD = hD = C Re nD Pr1 3, where k

T∞ , h

P = 2w + 2t Ac = w t t w

L Tb

ReD 1–4 4 – 40 40 – 4,000 4,000 – 40,000 40,000 – 250,000

C 0.989 0.911 0.683 0.193 0.0266

n 0.330 0.385 0.466 0.618 0.805

Flow Over a Sphere of Diameter, D Pin Fin P= π D

T∞ , h D

Ac =

πD 2 4

NuD = hD = 2.0 + 0.60 Re1D 2 Pr1 3, k ^1 < ReD< 70, 000; 0.6 < Pr < 400h

Internal Flow

ReD =

tumD n

L Tb

86

HEAT TRANSFER

Laminar Flow in Circular Tubes For laminar flow (ReD < 2300), fully developed conditions NuD = 4.36

(uniform heat flux)

NuD = 3.66

(constant surface temperature)

For laminar flow (ReD < 2300), combined entry length with constant surface temperature NuD = 1.86 f

1 3

ReDPr L p D

d

nb n ns

Outside Horizontal Tubes 0.25

tl2 gh fg D3 H NuD = hD = 0.729 > k nlkl _Tsat - Tsi

0.14

, where

D = tube outside diameter (m) Note: Evaluate all liquid properties at the average temperature between the saturated temperature, Tsat, and the surface temperature, Ts.

L = length of tube (m) D = tube diameter (m) μb = dynamic viscosity of fluid [kg/(s•m)] at bulk temperature of fluid, Tb μs = dynamic viscosity of fluid [kg/(s•m)] at inside surface temperature of the tube, Ts

Natural (Free) Convection Vertical Flat Plate in Large Body of Stationary Fluid Equation also can apply to vertical cylinder of sufficiently large diameter in large body of stationary fluid.

Turbulent Flow in Circular Tubes For turbulent flow (ReD > 104, Pr > 0.7) for either uniform surface temperature or uniform heat flux condition, SiederTate equation offers good approximation: n NuD = 0.023 Re0D.8 Pr1 3 d nb n s

, where

0.14

Non-Circular Ducts In place of the diameter, D, use the equivalent (hydraulic) diameter (DH) defined as DH = 4 # cross -sectional area wetted perimeter Circular Annulus (Do > Di) In place of the diameter, D, use the equivalent (hydraulic) diameter (DH) defined as

hr = C b k l RaLn, where L L

= the length of the plate (cylinder) in the vertical direction gb _Ts - T3i L3 RaL = Rayleigh Number = Pr v2 Ts = surface temperature (K) T∞ = fluid temperature (K) β = coefficient of thermal expansion (1/K) 2 with T in absolute temperature) (For an ideal gas: b = Ts + T3 υ

= kinematic viscosity (m2/s) Range of RaL 104 – 109 109 – 1013

DH = Do - Di

C 0.59 0.10

n 1/4 1/3

Liquid Metals (0.003 < Pr < 0.05) NuD = 6.3 + 0.0167 Re0D.85 Pr0.93 (uniform heat flux)

Long Horizontal Cylinder in Large Body of Stationary Fluid h = C b k l Ra nD, where D

NuD = 7.0 + 0.025 Re0D.8 Pr0.8 (constant wall temperature) Condensation of a Pure Vapor On a Vertical Surface

RaD =

gb _Ts - T3i D3 Pr v2

RaD 10 – 102 102 – 104 104 – 107 107 – 1012

C 1.02 0.850 0.480 0.125

0.25

ρl g hfg L μl kl Tsat Ts

tl2 gh fg L3 H NuL = hL = 0.943 > k nlkl _Tsat - Tsi

, where

= density of liquid phase of fluid (kg/m3) = gravitational acceleration (9.81 m/s2) = latent heat of vaporization [J/kg] = length of surface [m] = dynamic viscosity of liquid phase of fluid [kg/(s•m)] = thermal conductivity of liquid phase of fluid [W/(m•K)] = saturation temperature of fluid [K] = temperature of vertical surface [K]

Note: Evaluate all liquid properties at the average temperature between the saturated temperature, Tsat, and the surface temperature, Ts.

–3

n 0.148 0.188 0.250 0.333

Heat Exchangers The rate of heat transfer in a heat exchanger is Qo = UAFDTlm, where A F U ΔTlm

= any convenient reference area (m2) = heat exchanger configuration correction factor (F = 1 if temperature change of one fluid is negligible) = overall heat transfer coefficient based on area A and the log mean temperature difference [W/(m2•K)] = log mean temperature difference (K) HEAT TRANSFER

87

Heat Exchangers (cont.) Overall Heat Transfer Coefficient for Concentric Tube and Shell-and-Tube Heat Exchangers D ln d o n R R fo Di fi 1 1 1 , where UA = hiAi + Ai + 2rkL + Ao + hoAo Ai Ao Di Do hi ho k Rfi Rfo

2

= inside area of tubes (m ) = outside area of tubes (m2) = inside diameter of tubes (m) = outside diameter of tubes (m) = convection heat transfer coefficient for inside of tubes [W/(m2•K)] = convection heat transfer coefficient for outside of tubes [W/(m2•K)] = thermal conductivity of tube material [W/(m•K)] = fouling factor for inside of tube [(m2•K)/W] = fouling factor for outside of tube [(m2•K)/W]

Log Mean Temperature Difference (LMTD) For counterflow in tubular heat exchangers DTlm =

_THo - TCi i - _THi - TCoi

ln d

THo - TCi n THi - TCo

For parallel flow in tubular heat exchangers DTlm =

ΔTlm THi THo TCi TCo

_THo - TCoi - _THi - TCi i

T - TCo n ln d Ho THi - TCi

, where

= log mean temperature difference (K) = inlet temperature of the hot fluid (K) = outlet temperature of the hot fluid (K) = inlet temperature of the cold fluid (K) = outlet temperature of the cold fluid (K)

Heat Exchanger Effectiveness, ε Qo actual heat transfer rate f= o = maximum possible heat transfer rate Qmax f=

C _T - TCi i CH _THi - THoi or f = C Co Cmin _THi - TCi i Cmin _THi - TCi i

where C = mc o P = heat capacity rate (W/K) Cmin = smaller of CC or CH Number of Transfer Units (NTU) NTU = UA Cmin

88

HEAT TRANSFER

Effectiveness-NTU Relations C Cr = min = heat capacity ratio Cmax For parallel flow concentric tube heat exchanger f=

1 - exp 8- NTU ^1 + Cr hB 1 + Cr

NTU =-

ln 81 - f ^1 + Cr hB 1 + Cr

For counterflow concentric tube heat exchanger f=

1 - exp 8- NTU ^1 - Cr hB

1 - Crexp 8- NTU ^1 - Cr hB

f = NTU 1 + NTU NTU = 1 ln c f - 1 m Cr - 1 fCr - 1 NTU = f 1-f

^Cr< 1h ^Cr = 1h ^Cr< 1h

^Cr = 1h

RADIATION Types of Bodies Any Body For any body, α + ρ + τ = 1 , where α = absorptivity (ratio of energy absorbed to incident energy) ρ = reflectivity (ratio of energy reflected to incident energy) τ = transmissivity (ratio of energy transmitted to incident energy) Opaque Body For an opaque body: α + ρ = 1 Gray Body A gray body is one for which α = ε, (0 < α < 1; 0 < ε < 1), where ε = the emissivity of the body For a gray body: ε + ρ = 1 Real bodies are frequently approximated as gray bodies. Black body A black body is defined as one which absorbs all energy incident upon it. It also emits radiation at the maximum rate for a body of a particular size at a particular temperature. For such a body α=ε=1

Shape Factor (View Factor, Configuration Factor) Relations Reciprocity Relations

One-Dimensional Geometry with Thin Low-Emissivity Shield Inserted between Two Parallel Plates Radiation Shield

Q12

AiFij = AjFji, where Ai = surface area (m2) of surface i Fij = shape factor (view factor, configuration factor); fraction of the radiation leaving surface i that is intercepted by surface j; 0 ≤ Fij ≤ 1

ε3, 1

Summation Rule for N Surfaces

A1 , T1, ε1

N

! Fij = 1

j=1

Qo 12 =

Qo 12 = the net heat transfer rate from the body (W) ε = the emissivity of the body σ = the Stefan-Boltzmann constant [σ = 5.67 × 10-8 W/(m2•K4)] A = the body surface area (m2) T1 = the absolute temperature [K] of the body surface T2 = the absolute temperature [K] of the surroundings

A2 , T2 , ε2 A3 , T3

Net Energy Exchange by Radiation between Two Bodies Body Small Compared to its Surroundings Qo 12 = fvA `T14 - T24j, where

ε3, 2

v `T14 - T24j 1 - f3, 1 1 - f3, 2 1 - f1 1 - f2 1 1 f1A1 + A1F13 + f3, 1A3 + f3, 2A3 + A3F32 + f2A2

Reradiating Surface Reradiating Surfaces are considered to be insulated or adiabatic _Qo R = 0i .

A1 , T1 , ε1

Net Energy Exchange by Radiation between Two Black Bodies The net energy exchange by radiation between two black bodies that see each other is given by Qo 12 = A1F12 v `T14 - T24j

Q12

AR , TR , εR

A2 , T2 , ε2

Net Energy Exchange by Radiation between Two DiffuseGray Surfaces that Form an Enclosure Generalized Cases A1 , T1 , ε1

Qo 12 =

v `T14 - T24j 1 - f1 1 - f2 1 -1 + f A f1A1 + 2 2 A1F12 + =c 1 m + c 1 mG A1F1R A2F2R

A2 , T2 , ε2 Q12

Q12

A1 , T1 , ε1 A2 , T2 , ε2

Qo 12 =

v `T14 - T24j 1 - f1 1 - f2 1 f1A1 + A1F12 + f2A2

HEAT TRANSFER

89

TRANSPORT PHENOMENA MOMENTUM, HEAT, AND MASS TRANSFER ANALOGY

Rate of transfer as a function of gradients at the wall Momentum Transfer:

For the equations which apply to turbulent flow in circular tubes, the following definitions apply: Nu = Nusselt Number ; hD E k Pr = Prandtl Number (cpμ/k),

xw =-n c dv m =dy w

Dp m ftV 2 bDlc 8 = 4 - L f

Heat Transfer: o d Q n =-k c dT m A w dy w

Re = Reynolds Number (DVρ/μ), Sc = Schmidt Number [μ/(ρDm)], Sh = Sherwood Number (kmD/Dm),

Mass Transfer in Dilute Solutions:

St = Stanton Number [h/(cpG)],

dc b N l =-Dm d m n A w dy w

cm = concentration (mol/m3), cp = heat capacity of fluid [J/(kg•K)],

Rate of transfer in terms of coefficients Momentum Transfer: ftV 2 xw = 8

D = tube inside diameter (m), 2

Dm = diffusion coefficient (m /s), (dcm/dy)w = concentration gradient at the wall (mol/m4), (dT/dy)w = temperature gradient at the wall (K/m), (dv/dy)w f

Heat Transfer:

= velocity gradient at the wall (s–1),

o d Q n = hDT A w

= Moody friction factor,

G = mass velocity [kg/(m2•s)], h

= heat-transfer coefficient at the wall [W/(m2•K)],

k

= thermal conductivity of fluid [W/(m•K)],

Mass Transfer:

b N l = km Dcm A w

km = mass-transfer coefficient (m/s), L

= length over which pressure drop occurs (m),

(N/A)w = inward mass-transfer flux at the wall [mol/(m2•s)], _Qo Aiw = inward heat-transfer flux at the wall (W/m2),

y

= distance measured from inner wall toward centerline (m),

Δcm = concentration difference between wall and bulk fluid (mol/m3), ΔT = temperature difference between wall and bulk fluid (K), μ

2

= absolute dynamic viscosity (N•s/m ), and

τw = shear stress (momentum flux) at the tube wall (N/m ). 2

Definitions already introduced also apply.

90

TRANSPORT PHENOMENA

Use of friction factor (f ) to predict heat-transfer and masstransfer coefficients (turbulent flow) Heat Transfer: f jH = b Nu l Pr 2 3 = RePr 8 Mass Transfer: f jM = b Sh l Sc 2 3 = 8 ReSc

BIOLOGY For more information on Biology see the ENVIRONMENTAL ENGINEERING section. CELLULAR BIOLOGY ♦ PERIPLASMIC SPACE OUTER MEMBRANE

3 μm

1 μm SEX PILUS

RIBOSOMES PEPTIDOGLYCAN CYTOPLASMIC CHROMOSOME

FLAGELLA

COMMON PILUS

INNER OR CYTOPLASMIC MEMBRANE

10 μm

20 μm CELL WALL

STARCH GRANULES MITOCHONDRIA PLASMA MEMBRANE NUCLEUS ENDOPLASMIC RETICULUM

CHLOROPLAST

TONOPLAST

GOLGI COMPLEX

LYSOSOME

CENTRAL VACUOLE

ROUGH ENDOPLASMIC RETICULUM ANIMAL

PLANT

• Primary Subdivisions of Biological Organisms Group

Cell structure

Properties

Constituent groups

Eucaryotes

Eucaryotic

Multicellular; extensive differentiation of cells and tissues Unicellular, coenocytic or mycelial; little or no tissue differentiation

Plants (seed plants, ferns, mosses) Animals (vertebrates, invertebrates) Protists (algae, fungi, protozoa)

Eubacteria

Procaryotic

Cell chemistry similar to eucaryotes

Most bacteria

Archaebacteria

Procaryotic

Distinctive cell chemistry

Methanogens, halophiles, thermoacidophiles

♦ Shuler, Michael L., & Fikret Kargi, Bioprocess Engineering Basic Concepts, Prentice Hall PTR, New Jersey, 1992.

•

Stanier, Roger; Adelberg, Edward A; Wheelis, Mark L; Decastells; Painter, Page R; Ingraham, John L; The Microbial World, 5th ed., 1986. Reprinted by permission of Pearson Education, Inc., Upper Saddle River, NJ.

BIOLOGY

91

Biochemical Catabolic Pathways Catabolism is the breakdown of nutrients to obtain energy and precursors for biosynthesis. Carbohydrates are the most important class of carbonaceous nutrients for fermentations, although some microbial species can also utilize amino acids, hydrocarbons and other compounds. As an illustration of microbial diversity, almost any carbohydrate or related compound can be fermented by some microbe. Embden-Meyerhof-Parnas (EMP) 6

CH2 OH 5 H H O H 1 4 HO OH H OH

GLYCERALDEHYDE 3-PHOSPHATE GLUCOSE

GLYCERALDEHYDE 3-PHOSPHATE DEHYDROGENASE

2

3

H

OH ATP

HEXOKINASE

C

CH2 OPO 3 2 O H H H OH HO OH H OH

ADP

KINASE

ATP

TAKEN TWICE FOR EACH GLUCOSE MOLECULE

CH 2 OPO 32

CH 3OH FRUCTOSE 6-PHOSPHATE

H H HO OH OH H ATP PHOSPHOFRUCTOKINASE

ADP CH2 OPO 3 2

H

OPO 3

3-PHOSPHOGLYCERATE

GLUCOSE 6-PHOSPHATE

CH2 OPO 3 2

H HO

1,3-DIPHOSPHOGLYCERATE

2

O

PHOSPHOHEXOISOMERASE

O

NADH 2 CH 2 OPO 3

HCOH

ADP

O

NAD +

OH

3

CH 2 OPO 3

2

FRUCTOSE 1,6-DIPHOSPHATE

OH

OH H

HCOH

3-PHOSPHOGLYCERATE

COO PHOSPHOGLYCERAMUTASE

CH 2 OH HCOPO 3 2

2-PHOSPHOGLYCERATE

COO ENOLASE

H2O CH 2 C

2

O PO 3

COO

PHOSPHOENOLPYRUVATE

ADP PYRUVATE KINASE

ATP CH 3

ALDOLASE

CH 2 OPO 3

2

H C=O

C=O COO

TRIOSE ISOMERASE

C=O CH 2 OH DIHYDROXYACETONE PHOSPHATE

HCOH CH 2 OPO 3

PYRUVATE

2

GLYCERALDEHYDE 3-PHOSPHATE

The Embden-Meyerhof-Parnas (EMP) pathway. Notice that each six-carbon glucose substrate molecule yields two three-carbon intermediates, each of which passes through the reaction sequence on the right-hand side.

Bailey, J.E. and D.F. Olis. Biochemical Engineering Fundamentals, 2nd ed., McGraw-Hill, New York, 1986.

92

BIOLOGY

Overall Stoichiometry

C6H12O6 + Pi + 2ADP + 2NAD+ " 2C3H4O3 + 2ATP + 2 _ NADH + H+i

where ADP is Adenosine diphosphate ATP is Adenosine triphosphate. ADP and ATP are the primary energy carriers in biosynthesis. Each mole of ATP carries –7.3 kcal free energy in the process of being reduced to ADP. NAD is Nicotinamide adenine dinucleotide (+ denotes the oxidized form and NADH denotes the reduced form). The role of this group is to serve as an electron carrier. C3H4O3 is pyruvate, a compound of central importance in biosynthesis reactions. The EMP pathway, among several other biochemical pathways, provides carbon skeletons for cellular biosynthesis and provides energy sources via substrate level phosphorylation. The EMP pathway is perhaps the most common carbohydrate catabolic pathway. Other catabolic pathways include the pentose phosphate cycle, yielding 1.67 ATP for each mole of glucose and the Entner-Doudoroff (ED) pathway, yielding 1 ATP per mole of glucose. Pathways are called upon by the elaborate control mechanisms within the organism depending on needed synthesis materials and available nutrients. Biochemical Respiratory Pathways Respiration is an energy producing process in which organic or reduced inorganic compounds are oxidized by inorganic compounds. If oxygen is the oxidizer, the processes yields carbon dioxide and water and is denoted as aerobic. Otherwise the process is denoted as facultative or anaerobic. Respiratory processes also serve as producers of precursors for biosynthesis. A summary of reductants and oxidants in bacterial respirations are shown in the table below. TABLE 5.4 REDUCTANTS AND OXIDANTS IN BACTERIAL RESPIRATIONS REDUCTANT

OXIDANT

PRODUCTS

ORGANISM

H2 H2

O2 SO 24

H 2O H 2 O + S2

HYDROGEN BACTERIA Desulfovibrio

ORGANIC COMPOUNDS NH 3 NO 2

O2 O2 O2

CO 2 + H 2 O NO 2 + H 2 O NO 3 + H 2 O

MANY BACTERIA, ALL PLANTS AND ANIMALS NITRIFYING BACTERIA NITRIFYING BACTERIA

ORGANIC COMPOUNDS Fe 2 + 2 S

NO 3 O2 O2

N 2 + CO2 Fe 3 + SO 24 + H 2 O

DENITRIFYING BACTERIA Ferrobacillus (iron bacteria) Thiobacillus (sulfur bacteria)

From W.R. Sistrom, Microbial Life, 2d ed., table 4-2, p. 53, Holt, Rinehart, and Winston, Inc. New York, 1969.

BIOLOGY

93

The most important of the respiratory cycles is the tricarboxylic acid (denoted as TCA, citric acid cycle or Krebs cycle). The TCA cycle is shown in the figure below.

O CH 3 CCOOH Pyruvic acid CoASH CO2

2H (to NAD)

O

CH 3 C CoA 2H COOH Acetyl-coenzyme A COOH (to NAD) HCOH C O COOH Oxaloacetic acid C 4

COOH HO

C

CH 2

CH 2

H2 O

HSCoA

COOH Malic acid C 4 H2 O

CH 2 COOH

COOH

CH 2

CH

COOH Citric acid C 6

CH COOH Fumaric Acid C 4

COOH HC

CH 2 COOH

CHOH COOH Isocitric acid C 6 2H (to NAD) CO2

TCA (tricarboxylic acid, Krebs, citric acid) CYCLE

2H (to FAD) COOH CH 2 CH 2

COOH Succinic acid C 4 CoASH

H 2 C CH 2 COOH C O COOH α− Ketoglutaric acid C 5

ATP

COOH

ATP + P1

CH 2 CoASH

CO 2 CH 2 2H (to NAD) C-SCoA

H2 O

O Succinyl CoA The tricarboxylic acid cycle Bailey, J.E. and D.F. Olis, Biochemical Engineering Fundamentals, 2nd ed., McGraw-Hill, New York, 1986.

94

BIOLOGY

Adding the EMP reactions with the TCA and oxidative phosphorylation reactions (not shown), results in a stoichiometry giving a net upper bound on ATP yield from glucose in a respiring cell with glucose as the primary carbon source and oxygen as the electron donor. C6H12O6 + 38ADP + 38Pi + 6O2 " 6CO2 + 38ATP + 44H2O The free energy is approximately 38 moles (–7.3 kcal/mole) or –277 kcal/mole glucose. Anaerobic reactions, where the carbon source and/or electron donor may be other inorganic or organic substrates, provide less free energy and thus result in lower product yields and lower biomass production. The energy capture efficiency of the EMP-TCA sequence compared to the inorganic combustion reaction of glucose (net free energy of –686 kcal/mole glucose with the negative sign indicating energy evolved) is about 40%. Photosynthesis Photosynthesis is a most important process form synthesizing glucose from carbon dioxide. It also produces oxygen. The most important photosynthesis reaction is summarized as follows. 6CO2 + 6H2O + light " C6H12O6 + 6O2 The light is required to be in the 400- to 700-nm range (visible light). Chlorophyll is the primary photosynthesis compound and it is found in organisms ranging from tree and plant leaves to single celled algae. The reaction above requires an energy equivalent of roughly 1,968 kcal per mole of glucose produced. The inorganic synthesis of glucose requires +686 kcal/mole glucose produced. Hence the efficiency of photosynthesis is roughly (686 kcal/mole glucose inorganic)/(1,968 kcal/mole glucose photosynthesis) 686 kcal/mole or 35%. 1968 kcal/mole

♦ Organismal Growth in Batch Culture 106

BACTERIAL NUMBERS

An associated respiratory reaction is the oxidative phosphorylation in which ATP is regenerated.

STATIONARY PHASE

105

DEATH PHASE

LOG GROWTH PHASE

104 LAG PHASE

ACCELERATED GROWTH PHASE

103 5

10

15

20

25 30 35 TIME (h)

40

45

50

55

60

Exponential (log) growth with constant specific growth rate, μ n = b 1x l b dx l, where dt x = the cell/organism number or cell/organism concentration t = time (hr) μ = the specific growth rate (time–1) while in the exponential growth phase. Logistic Growth–Batch Growth including initial into stationary phase dx = kx c1 - x m x3 dt x x = x 0 _1 - e kt i, where 3 where, k = logistic growth constant (h–1), x0 = initial concentration (g/l) x∞ = carrying capacity (g/l).

Other bacterial photosynthesis reactions may occur where the carbon source is other organic or inorganic substrates.

♦ Davis, M.L., Principles of Environmental Engineering, McGraw-Hill, New York, 2004.

BIOLOGY

95

Characteristics of Selected Microbial Cells Organism genus or type

Gram reaction2

Metabolism1

Type

Morphological characteristics3

Escherichia

Bacteria

Chemoorganotroph-facultative

Negative

Rod–may or may not be motile, variable extracellular material

Enterobacter

Bacteria


Negative

Rod–motile; significant extracellular material

Bacillus

Bacteria

Chemoorganotroph-aerobic

Positive

Rod–usually motile; spore; can be significant extracellular material

Lactobacillus

Bacteria


Variable

Rod–chains–usually nonmotile; little extracellular material

Staphylococcus

Bacteria


Positive

Cocci–nonmotile; moderate extracellular material

Nitrobacter

Bacteria

Chemoautotroph-aerobic; can use nitrite as electron donor

Negative

Short rod–usually nonmotile; little extracellular material

Rhizobium

Bacteria

Chemoorganotroph-aerobic; nitrogen fixing

Negative

Rods–motile; copious extracellular slime

Pseudomonas

Bacteria

Chemoorganotroph-aerobic and some chemolithotroph facultative (using NO3 as electron acceptor)

Negative

Rods–motile; little extracellular slime

Thiobacillus

Bacteria

Chemoautotroph-facultative

Negative

Rods–motile; little extracellular slime

Clostridium

Bacteria

Chemoorganotroph-anaerobic

Positive

Rods–usually motile; spore; some extracellular slime

Methanobacterium

Bacteria

Chemoautotroph-anaerobic

Unknown

Rods or cocci–motility unknown; some extracellular slime

Chromatium

Bacteria

Photoautotroph-anaerobic

N/A

Rods–motile; some extracellular material

Spirogyra

Alga

Photoautotroph-aerobic

N/A

Rod/filaments; little extracellular material

Aspergillus

Mold

Chemoorganotroph-aerobic and facultative

--

Filamentous fanlike or cylindrical conidia and various spores

Candida

Yeast

Chemoorganotroph-aerobic and facultative

--

Usually oval but can form elongated cells, mycelia, and various spores

Saccharomyces

Yeast


--

Spherical or ellipsoidal; reproduced by budding; can form various spores

1

Aerobic – requires or can use oxygen as an electron receptor. Facultative – can vary the electron receptor from oxygen to organic materials. Anaerobic – organic or inorganics other than oxygen serve as electron acceptor. Chemoorganotrophs – derive energy and carbon from organic materials. Chemoautotrophs – derive energy from organic carbons and carbon from carbon dioxide. Some species can also derive energy from inorganic sources. Photolithotrophs – derive energy from light and carbon from CO2. May be aerobic or anaerobic.

2

Gram negative indicates a complex cell wall with a lipopolysaccharide outer layer. Gram positive indicates a less complicated cell wall with a peptide-based outer layer.

3

Extracellular material production usually increases with reduced oxygen levels (e.g., facultative). Carbon source also affects production; extracellular material may be polysaccharides and/or proteins; statements above are to be understood as general in nature.

Adapted from Pelczar, M.J., R.D. Reid, and E.C.S. Chan, Microbiology: Concepts and Applications, McGraw-Hill, New York, 1993.

96

BIOLOGY

Transfer Across Membrane Barriers Mechanisms Passive diffusion – affected by lipid solubility (high solubility increases transport), molecular size (decreased with molecular size), and ionization (decreased with ionization). Passive diffusion is influenced by: 1. Partition coefficient (indicates lipid solubility; high lipid solubility characterizes materials that easily penetrate skin and other membranes). 2. Molecular size is important in that small molecules tend to transport much easier than do large molecules. 3. Degree of ionization is important because, in most cases, only unionized forms of materials transport easily through membranes. Ionization is described by the following relationships: Acids

pKa - pH = log10 ; nonionized form E = log10 HA A ionized form

Base + pKa - pH = log10 ; ionized form E = log10 HB B nonionized form

Facilitated diffusion – requires participation of a protein carrier molecule. This mode of transport is highly compound dependent. Active diffusion – requires protein carrier and energy and is similarly affected by ionization and is highly compound dependent. Other – includes the specialized mechanisms occurring in lungs, liver, and spleen.

BIOLOGY

97

BIOPROCESSING Stoichiometry of Selected Biological Systems Aerobic Production of Biomass and a Single Extracellular Product CNcsHm*NcsOn*Ncs + aO2 + bNH3 → cCNccHα*NccOβ*NccNδ*Ncc + dCNcpHx*NcpOy*NcpNz*Ncp + eH2O + fCO2 Substrate

Biomass

Bioproduct

where Ncs = the number of carbons in the substrate equivalent molecule Ncc = the number of carbons in the biomass equivalent molecule Ncp = the number of carbons in the bioproduct equivalent molecule Coefficients m, n, α, β, x, y, z as shown above are multipliers useful in energy balances. The coefficient times the respective number of carbons is the equivalent number of constituent atoms in the representative molecule as shown in the following table. Degrees of Reduction (available electrons per unit of carbon) γs = 4 + m – 2n γb = 4 + α – 2β – 3δ γp = 4 + x – 2y – 3z Subscripts refer to substrate (s), biomass (b), or product (p). A high degree of reduction denotes a low degree of oxidation. Carbon balance cNcc + dNcp + f = Ncs Nitrogen balance cδNcc + dzNcp = b Electron balance cγb Ncb + dγp Ncp = γs Ncs– 4a Energy balance Qocγb Ncc + Qodγp Ncp = Qoγs Ncs – Qo4a, Qo = heat evolved per unit of equivalent of available electrons ≈ 26.95 kcal/mole O2 consumed Respiratory quotient (RQ) is the CO2 produced per unit of O2 f RQ = a Yield coefficient = c (grams of cells per gram substrate, YX/S) or = d (grams of product per gram substrate, YX/XP) Satisfying the carbon, nitrogen, and electron balances plus knowledge of the respiratory coefficient and a yield coefficient is sufficient to solve for a, b, c, d, and f coefficients.

98

BIOLOGY

Composition Data for Biomass and Selected Organic Compounds ♦ COMPOUND BIOMASS

DEGREE OF

MOLECULAR FORMULA CH1 64N0 16O0,52 P0.0054S0.005a

Energy balance QocγbNcc = QoγsNcs – Qo4a Qo ≈ 26.95 kcal/mole Ο2 consumed

REDUCTION, γ

WEIGHT, m

4.17 (NH3)

24.5

Aerobic Biodegradation of Glucose with No Product, Ammonia Nitrogen Source, Cell Production Only, RQ = 1.1 C6H12O6 + aO2 + bNH3 → cCH1.8O0.5N0.2 + dCO2 + eH2O Substrate Cells

4.65 (N3) 5.45 (HNO3)

METHANE

CH4

8

16.0

n-ALKANE

C4H32

6.13

14.1

METHANOL

CH4O

6.0

32.0

ETHANOL

C2H6O

6.0

23.0

GLYCEROL

C2H6O3

4.67

30.7

MANNITOL

C6H14O6

4.33

30.3

ACETIC ACID

C2H4O2

4.0

30.0

LACTIC ACID

C3H6O3

4.0

30.0

GLUCOSE

C6H12O6

4.0

30.0

FORMALDEHYDE

CH2O

4.0

30.0

GLUCONIC ACID

C6H12O7

3.67

32.7

SUCCINIC ACID

C4H6O4

3.50

29.5

CITRIC ACID

C6H8O7

3.0

32.0

MALIC ACID

C4H6O5

3.0

33.5

FORMIC ACID

CH2O2

2.0

46.0

OXALIC ACID

C2H2O4

1.0

45.0

For the above conditions, one finds that: a = 1.94 b = 0.77 c = 3.88 d = 2.13 e = 3.68 The c coefficient represents a theoretical maximum yield coefficient, which may be reduced by a yield factor. Anaerobic Biodegradation of Organic Wastes, Incomplete Stabilization CaHbOcNd → nCwHxOyNz + mCH4 + sCO2 + rH2O + (d – nx)NH3

The weights in column 4 of the above table are molecular weights expressed as a molecular weight per unit carbon; frequently denoted as reduced molecular weight. For example, glucose has a molecular weight of 180 g/mole and 6 carbons for a reduced molecular weight of 180/6 = 30.

s = a – nw – m r = c – ny – 2s

Except for biomass, complete formulas for compounds are given in column 2 of the above table. Biomass may be represented by:

Anaerobic Biodegradation of Organic Wastes, Complete Stabilization

C4.4H7.216N0.704 O2.288P0.0237S0.022 Aerobic conversion coefficients for an experimental bioconversion of a glucose substrate to oxalic acid with indicated nitrogen sources and conversion ratios with biomass and one product produced.

Knowledge of product composition, yield coefficient (n) and a methane/CO2 ratio is needed.

CaHbOcNd + rH2O → mCH4 + sCO2 + dNH3 r = 4 a - b - 2 c + 3d 4 a b 4 2c + 3 d + s= 8

m = 4 a + b - 2 c - 3d 8

Nitrogen Source Theoretical yield coefficientsa

NH3 (shown in the above equation)

HNO3

N2

a

2.909

3.569

3.929

b

0.48

0.288

0.144

c

0.6818

0.409

0.409

d

0.00167

0.00167

0.00167

e

4.258

4.66

4.522

f

2.997

4.197

4.917

Substrate to cells conversion ratio

0.5

0.3

0.3

Substrate to bioproduct conversion ratio

0.1

0.1

0.1

Respiration Quotient (RQ)

1.03

1.17

1.06

a The stoichiometric coefficients represent theoretical balance conditions. Product yields in practice are frequently lower and may be adjusted by the application of yield coefficients.

♦ B. Atkinson and F. Mavitona, Biochemical Engineering and Biotechnology Handbook, Macmillan, Inc., 1983. Used with permission of Nature Publishing Group (www.nature.com).

BIOLOGY

99

CHEMISTRY Avogadro’s Number: The number of elementary particles in a mol of a substance. 1 mol = 1 gram mole 1 mol = 6.02 × 1023 particles A mol is defined as an amount of a substance that contains as many particles as 12 grams of 12C (carbon 12). The elementary particles may be atoms, molecules, ions, or electrons.

ACIDS, BASES, and pH (aqueous solutions) pH = log10 e 1 o , where 7 H+A

Equilibrium Constant of a Chemical Reaction aA + bB E cC + dD c d 6 @ 6 @ Keq = C a D b 6 A@ 6 B @

Le Chatelier's Principle for Chemical Equilibrium – When a stress (such as a change in concentration, pressure, or temperature) is applied to a system in equilibrium, the equilibrium shifts in such a way that tends to relieve the stress.

[H+] = molar concentration of hydrogen ion, in gram moles per liter

Heats of Reaction, Solution, Formation, and Combustion – Chemical processes generally involve the absorption or evolution of heat. In an endothermic process, heat is absorbed (enthalpy change is positive). In an exothermic process, heat is evolved (enthalpy change is negative).

Acids have pH < 7.

Solubility Product of a slightly soluble substance AB:

Bases have pH > 7.

AmBn → mAn+ + nBm–

ELECTROCHEMISTRY

Solubility Product Constant = KSP = [A+]m [B–]n

Cathode – The electrode at which reduction occurs.

Metallic Elements – In general, metallic elements are distinguished from nonmetallic elements by their luster, malleability, conductivity, and usual ability to form positive ions.

Anode – The electrode at which oxidation occurs. Oxidation – The loss of electrons. Reduction – The gaining of electrons. Oxidizing Agent – A species that causes others to become oxidized. Reducing Agent – A species that causes others to be reduced. Cation – Positive ion Anion – Negative ion

DEFINITIONS Molarity of Solutions – The number of gram moles of a substance dissolved in a liter of solution. Molality of Solutions – The number of gram moles of a substance per 1,000 grams of solvent. Normality of Solutions – The product of the molarity of a solution and the number of valence changes taking place in a reaction. Equivalent Mass – The number of parts by mass of an element or compound which will combine with or replace directly or indirectly 1.008 parts by mass of hydrogen, 8.000 parts of oxygen, or the equivalent mass of any other element or compound. For all elements, the atomic mass is the product of the equivalent mass and the valence. Molar Volume of an Ideal Gas [at 0°C (32°F) and 1 atm (14.7 psia)]; 22.4 L/(g mole) [359 ft3/(lb mole)]. Mole Fraction of a Substance – The ratio of the number of moles of a substance to the total moles present in a mixture of substances. Mixture may be a solid, a liquid solution, or a gas.

100

CHEMISTRY

Nonmetallic Elements – In general, nonmetallic elements are not malleable, have low electrical conductivity, and rarely form positive ions. Faraday’s Law – In the process of electrolytic changes, equal quantities of electricity charge or discharge equivalent quantities of ions at each electrode. One gram equivalent weight of matter is chemically altered at each electrode for 96,485 coulombs, or one Faraday, of electricity passed through the electrolyte. A catalyst is a substance that alters the rate of a chemical reaction and may be recovered unaltered in nature and amount at the end of the reaction. The catalyst does not affect the position of equilibrium of a reversible reaction. The atomic number is the number of protons in the atomic nucleus. The atomic number is the essential feature which distinguishes one element from another and determines the position of the element in the periodic table. Boiling Point Elevation – The presence of a nonvolatile solute in a solvent raises the boiling point of the resulting solution compared to the pure solvent; i.e., to achieve a given vapor pressure, the temperature of the solution must be higher than that of the pure substance. Freezing Point Depression – The presence of a solute lowers the freezing point of the resulting solution compared to that of the pure solvent.

CHEMISTRY

101

21

39

Y

88.906

57*

40.078

38

Sr

87.62

K

39.098

37

Rb

85.468

22

104

Rf

138.91

89**

137.33

88

87

**Actinide Series

*Lanthanide Series

Ac

227.03

Ra

226.02

Fr

(223)

72

23

24

25

183.85

180.95

93 Np 237.05

92 U 238.03

91 Pa 231.04

90

Th

232.04

(145)

144.24

Pm

140.91

Nd

140.12

59 Pr

Ce

58

Ha (262) 61

186.21

W

60

75 Re

74

73 Ta 105

(98)

Tc

43

54.938

Mn

95.94

Mo

42

51.996

Cr

92.906

Nb

41

50.941

V

(261)

178.49

Hf

132.91

La

56

Ba

55

91.224

Zr

40

47.88

Ti

Cs

44.956

Sc

20

Ca

19

24.305

22.990 26

27

29

30

31

32

33

97 Bk (247)

96 Cm (247)

95 Am (243)

94 Pu (244)

158.92

Tb 157.25

Gd 151.96

Eu

150.36

Sm

65

(251)

Cf

98

162.50

Dy

66

68

69

70

71

Lr (260)

No (259)

Md (258)

Fm (257) (252)

103

102

101 100 99 Es

174.97

Lu 173.04

Yb 168.93

Tm

(210)

At

85

126.90

I

53

79.904

Br

35

35.453

Cl

17

18.998

167.26

Er

9 F

164.93

Ho

67

208.98 207.2 204.38 200.59 196.97

195.08

192.22

190.2

64

(209)

Bi Tl

Pt

Ir

63

84 Po

83 82 Pb

81 80 Hg

79 Au

78

77

76 Os

62

127.60

Te

52

78.96

Se

34

121.75

Sb

51

74.921

As

118.71

Sn

50

72.61

Ge

114.82

In

49

69.723

Ga

112.41

Cd

48

65.39

Zn

32.066

S

16

15.999

107.87

Ag

47

63.546

Cu

30.974

P

15

14.007

8 O

106.42

Pd

46

58.69

Ni

28

28.086

Si

14

12.011

7 N

102.91

Rh

45

58.933

Co

26.981

Al

6 C

101.07

Ru

44

55.847

Fe

13

12

Mg

11

Na

10.811

9.0122

6.941

5 B

Be

Li

Atomic Weight

4

Symbol

Atomic Number

3

1.0079

H

1

PERIODIC TABLE OF ELEMENTS

(222)

Rn

86

131.29

Xe

54

83.80

Kr

36

39.948

Ar

18

20.179

Ne

10

4.0026

He

2

102

CHEMISTRY

Ethane

RH

Common Name

General Formula

Functional Group

Ethane

IUPAC Name

bonds

C–C

and

C–H

CH3CH3

Specific Example

Alkane

RC ≡ CR

R2C = CHR

C=C

R2C = CR2

–C ≡C–

RC ≡ CH

RCH = CHR

RCH = CH2

Acetylene

Acetylene

Ethylene

Ethylene

or

Ethyne

Ethene

or

HC ≡ CH

Alkyne

H2C = CH2

Alkene

Aromatic Ring

ArH

Benzene

Benzene

Arene

C

RX

X

Ethyl chloride

Chloroethane

CH3CH2Cl

Haloalkane

C

OH

ROH

Ethyl alcohol

Ethanol

CH3CH2OH

Alcohol

FAMILY

C

O C

ROR

Dimethyl ether

Methoxymethane

CH3OCH3

Ether

C

N

R3N

R2NH

RNH2

Methylamine

Methanamine

CH3NH2

Amine

IMPORTANT FAMILIES OF ORGANIC COMPOUNDS

CH3CCH3

CH3CH

C

O

RCH

O

H

Acetaldehyde

C

O

R1CR2

O

Dimethyl ketone

Acetone

O

O

Ethanal

Ketone

Aldehyde

CH3COCH3

CH3COH

C

O OH

RCOH

O

Acetic Acid

C

O O

RCOR

O

Methyl acetate

C

ethanoate

Methyl

O

O

Ethanoic Acid

Ester

Carboxylic Acid

Standard Oxidation Potentials for Corrosion Reactions* Corrosion Reaction

Potential, Eo, Volts vs. Normal Hydrogen Electrode

Au → Au3+ + 3e – 2H2O → O2 + 4H+ + 4e – Pt → Pt2+ + 2e – Pd → Pd2+ + 2e – Ag → Ag+ + e –

–1.498 –1.229 –1.200 –0.987 –0.799

2Hg → Hg22+ + 2e– Fe2+ → Fe3+ + e– 4(OH)– → O2 + 2H2O + 4e – Cu → Cu2+ + 2e– Sn2+ → Sn4+ + 2e –

–0.788 –0.771 –0.401 –0.337 –0.150

H2 → 2H+ + 2e – Pb → Pb2+ + 2e – Sn → Sn2+ + 2e – Ni → Ni2+ + 2e – Co → Co2+ + 2e–

0.000 +0.126 +0.136 +0.250 +0.277

Cd → Cd2+ + 2e – Fe → Fe2+ + 2e– Cr → Cr3+ + 3e– Zn → Zn2+ + 2e– Al → Al3+ + 3e –

+0.403 +0.440 +0.744 +0.763 +1.662

Mg → Mg2+ + 2e– Na → Na+ + e – K → K+ + e –

+2.363 +2.714 +2.925

* Measured at 25oC. Reactions are written as anode half-cells. Arrows are reversed for cathode half-cells. Flinn, Richard A. and Trojan, Paul K., Engineering Materials and Their Applications, 4th ed., Houghton Mifflin Company, 1990.

NOTE: In some chemistry texts, the reactions and the signs of the values (in this table) are reversed; for example, the half-cell potential of zinc is given as –0.763 volt for the reaction Zn2+ + 2e–→ Zn. When the potential Eo is positive, the reaction proceeds spontaneously as written.

CHEMISTRY

103

MATERIALS SCIENCE/STRUCTURE OF MATTER ATOMIC BONDING

TESTING METHODS

Primary Bonds Ionic (e.g., salts, metal oxides) Covalent (e.g., within polymer molecules) Metallic (e.g., metals)

Standard Tensile Test Using the standard tensile test, one can determine elastic modulus, yield strength, ultimate tensile strength, and ductility (% elongation). (See Mechanics of Materials section.)

CORROSION A table listing the standard electromotive potentials of metals is shown on the previous page. For corrosion to occur, there must be an anode and a cathode in electrical contact in the presence of an electrolyte.

Anode Reaction (Oxidation) of a Typical Metal, M Mo → Mn+ + ne– Possible Cathode Reactions (Reduction) ½ O2 + 2 e– + H2O → 2 OH– ½ O2 + 2 e– + 2 H3O+ → 3 H2O 2 e– + 2 H3O+ → 2 H2O + H2 When dissimilar metals are in contact, the more electropositive one becomes the anode in a corrosion cell. Different regions of carbon steel can also result in a corrosion reaction: e.g., cold-worked regions are anodic to noncoldworked; different oxygen concentrations can cause oxygen-deficient regions to become cathodic to oxygen-rich regions; grain boundary regions are anodic to bulk grain; in multiphase alloys, various phases may not have the same galvanic potential.

DIFFUSION

Endurance Test Endurance tests (fatigue tests to find endurance limit) apply a cyclical loading of constant maximum amplitude. The plot (usually semi-log or log-log) of the maximum stress (σ) and the number (N) of cycles to failure is known as an S-N plot. The figure below is typical of steel but may not be true for other metals; i.e., aluminum alloys, etc.

σ ENDURANCE LIMIT KNEE

LOG N (CYCLES)

The endurance stress (endurance limit or fatigue limit) is the maximum stress which can be repeated indefinitely without causing failure. The fatigue life is the number of cycles required to cause failure for a given stress level. Impact Test The Charpy Impact Test is used to find energy required to fracture and to identify ductile to brittle transition.

Diffusion Coefficient D = Do e−Q/(RT), where D = diffusion coefficient, Do = proportionality constant, Q = activation energy, R = gas constant [8.314 J/(mol•K)], and T = absolute temperature.

THERMAL AND MECHANICAL PROCESSING Cold working (plastically deforming) a metal increases strength and lowers ductility. Raising temperature causes (1) recovery (stress relief), (2) recrystallization, and (3) grain growth. Hot working allows these processes to occur simultaneously with deformation. Quenching is rapid cooling from elevated temperature, preventing the formation of equilibrium phases. In steels, quenching austenite [FCC (γ) iron] can result in martensite instead of equilibrium phases—ferrite [BCC (α) iron] and cementite (iron carbide).

104

MATERIALS SCIENCE/STRUCTURE OF MATTER

Impact tests determine the amount of energy required to cause failure in standardized test samples. The tests are repeated over a range of temperatures to determine the ductile to brittle transition temperature.

Creep Creep occurs under load at elevated temperatures. The general equation describing creep is: df = Av ne- Q dt

]RT g

where: ε = strain, t = time, A = pre-exponential constant, σ = applied stress, n = stress sensitivity. For polymers below, the glass transition temperature, Tg, n is typically between 2 and 4, and Q is ≥ 100 kJ/mol. Above Tg, n is typically between 6 and 10, and Q is ~ 30 kJ/mol. For metals and ceramics, n is typically between 3 and 10, and Q is between 80 and 200 kJ/mol.

Representative Values of Fracture Toughness Material A1 2014-T651 A1 2024-T3 52100 Steel 4340 Steel Alumina Silicon Carbide

K Ic (MPa•m 1/2 ) 24.2 44 14.3 46 4.5 3.5

K Ic (ksi•in 1/2 ) 22 40 13 42 4.1 3.2

HARDENABILITY OF STEELS Hardenability is the “ease” with which hardness may be attained. Hardness is a measure of resistance to plastic deformation. ♦

STRESS CONCENTRATION IN BRITTLE MATERIALS When a crack is present in a material loaded in tension, the stress is intensified in the vicinity of the crack tip. This phenomenon can cause significant loss in overall ability of a member to support a tensile load. KI = yv ra KI = the stress intensity in tension, MPa•m1/2, y = is a geometric parameter, y = 1 for interior crack (#2) and (#8) indicate grain size

y = 1.1 for exterior crack σ = is the nominal applied stress, and

JOMINY HARDENABILITY CURVES FOR SIX STEELS

a = is crack length as shown in the two diagrams below. ♦

a

EXTERIOR CRACK (y = 1.1)

2a

INTERIOR CRACK (y = 1)

The critical value of stress intensity at which catastrophic crack propagation occurs, KIc, is a material property.

COOLING RATES FOR BARS QUENCHED IN AGITATED WATER ♦Van Vlack,

L., Elements of Materials Science & Engineering, Addison-Wesley, Boston, 1989.

MATERIALS SCIENCE/STRUCTURE OF MATTER

105

♦

Also, for axially oriented, long, fiber-reinforced composites, the strains of the two components are equal. (ΔL/L)1 = (ΔL/L)2 ΔL = change in length of the composite, L

= original length of the composite. Material

RELATIONSHIP BETWEEN HARDNESS AND TENSILE STRENGTH For steels, there is a general relationship between Brinell hardness and tensile strength as follows: TS _ psii - 500 BHN

TS ^MPah - 3.5 BHN

ASTM GRAIN SIZE SV = 2PL

^

h

N_0.0645 mm2i = 2 n - 1

Nactual N , where = Actual Area _0.0645 mm 2i SV = PL = N = n =

grain-boundary surface per unit volume, number of points of intersection per unit length between the line and the boundaries, number of grains observed in a area of 0.0645 mm2, and grain size (nearest integer > 1).

COMPOSITE MATERIALS tc = Rfi ti Cc = Rfici f > z

BEARING PRESSURE qS

0.24

n = 2.0

L 0.22

0.250 0.248

n =1.8

0.240 0.237 0.227

n =1 .6

B

Z

n = 4 .0

n =1.4 At Point A: Δp = I Z q S

A 0.20

0.218 0.212

n = 1.2 n = 1.0

0.205

n = 0 .9

0.196

n = 0.8

0.185

n = 0 .7

0.172

n =0.6

0.156

n = 0.50

0.137

n = 0.45

0.127

n = 0.40

0.115

n = 0.35

0.103

n = 0.30

0.090

n = 0.25

0.076

n = 0.20

0.062

n = 0.15

0.047

n = 0.10

0.032

n = 0.05

0.016

for m = B / Z and n = L / Z

STRESS INFLUENCE VALUE, IZ

0.18

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.00 0.1

0.2

0.3

0.5

1.0

2.0

NORMALIZED BEARING WIDTH, m

140

CIVIL ENGINEERING

3.0

5.0

10.0

PERCENTAGE OF EXCESS PORE PRESSURE REMAINING = ue(t) / ue(t=0) 100

90

80

70

60

50

40

30

20

10

0

0.0 0.1 0.2 0.3

0.05

0.4

0.10

0.5

5

0.1

0.6

0

0.2

1.00

0.70

0.80 0.90

1.0

0.60

0.50

0.4

0.9

0.45

0 0.35

0.8

0.3

5

0

0.7

0.2

NORMALIZED POSITION WITHIN CONSOLIDATING LAYER = z / HDR

PORE PRESSURE DISTRIBUTION IN A CLAY LAYER BOUNDED AT TOP AND BOTTOM BY PERMEABLE LAYERS

1.1 1.2 1.3 1.4 1.5 1.6 1.7

CONTOURS OF TIME FACTOR, TV, ARE SHOWN CONTOUR INTERVAL FOR TV = 0.05

1.8 1.9 2.0 0

10

20

30

40

50

60

70

80

90

100

PERCENTAGE OF EXCESS PORE PRESSURE DISSIPATED = UV% = ( ue(t=0) – ue(t) ) / ue(t=0)

DEGREE OF CONSOLIDATION CURVE FOR PORE PRESSURE DISSIPATION WITH TIME

AVERAGE DEGREE OF CONSOLIDATION, UAV (%)

0.0

TV 0.00785 0.0314 0.0707 0.1000 0.1260 0.1960 0.2000 0.2860 0.3000 0.4000 0.4030 0.5000 0.5670 0.6000 0.7000 0.8000 0.8480 0.9000 1.0000 1.5000

20.0

40.0

60.0

80.0

UAV(%) 10.0 20.0 30.0 35.7 40.0 50.0 50.4 60.0 61.3 69.8 70.0 76.4 80.0 81.6 85.6 88.7 90.0 91.2 93.1 98.0

100.0 0.00

0.40

0.80

1.20

1.60

2.00

TIME FACTOR, TV

CIVIL ENGINEERING

141

= b FL l for bar force F caused by external load AE i

STRUCTURAL ANALYSIS Influence Lines for Beams and Trusses

= αLi(ΔT)i for temperature change in member

An influence line shows the variation of an effect (reaction, shear and moment in beams, bar force in a truss) caused by moving a unit load across the structure. An influence line is used to determine the position of a moveable set of loads that causes the maximum value of the effect. Moving Concentrated Load Sets R

P1

P2

= member misfit L, A

= member length and cross-sectional area

E

= member elastic modulus

Frame Deflection by Unit Load Method The displacement of any point on a frame caused by external loads is found by applying a unit load at that point that corresponds to the desired displacement:

Pn

e e

(α = coefficient of thermal expansion)

members

L/2

D=

L

The absolute maximum moment produced in a beam by a set of "n" moving loads occurs when the resultant "R" of the load set and an adjacent load are equal distance from the centerline of the beam. In general, two possible load set positions must be considered, one for each adjacent load. Beam Stiffness and Moment Carryover

θ

M B = M A /2 I

!

i=1

where: Δ

Li miMi # xx = = 0 EI dx i

= displacement at point of application of unit load (+ in direction of unit load)

mi = moment equation in member "i" caused by the unit load Mi = moment equation in member "i" caused by loads applied to frame Li = length of member "i" Ii

= moment of inertia of member "i"

If either the real loads or the unit load cause no moment in a member, that member can be omitted from the summation.

MA L

θ=

M L 4 EI

M =

kAB = stiffness

4 EI θ = kAB θ L

Member Fixed-End Moments (magnitudes)

w

MB = MA/2 = carryover A

B

Truss Deflection by Unit Load Method The displacement of a truss joint caused by external effects (truss loads, member temperature change, member misfit) is found by applying a unit load at the point that corresponds to the desired displacement. D joint = where: Δjoint fi (ΔL)i

142

FEMAB =

P

! fi ^DLhi

a

i=1

= force in member "i" caused by unit load (+ tension) = change in length caused by external effect (+ for increase in member length):

wL2 12

FEMBA =

members

= joint displacement at point of application of unit load ( + in direction of unit load )

CIVIL ENGINEERING

L

b

A

B L

FEMAB =

Pab L2

2

FEMBA =

P a2 b L2

STRUCTURAL DESIGN Definitions (based on ASCE 7-05) Allowable Stress Design: Method of proportioning structural members such that stresses produced in them by nominal loads do not exceed specified allowable stresses. Dead Load: Weights of all materials used in a building, including built-in partitions and fixed service equipment ("permanent load"). Design Strength (φRn): The product of the nominal strength and a resistance factor. Environmental Loads: Loads resulting from acts of nature, such as wind, snow, earthquake (each is a "transient load"). Factored Load (λQn): The product of the nominal load and a load factor. Limit State: A condition beyond which a structure or member becomes unfit for service because it is unsafe (strength limit state) or no longer performs its intended function (serviceability limit state). Live Load: All loads resulting from the use and occupancy of a building; does not include construction or other roof live loads, and environmental loads such as caused by wind, snow, earthquake, etc. (live loads are "transient loads"). Load Factor (λ): A factor that accounts for deviations of the actual load from the nominal load, or inaccuracies in the load models; also accounts for the probability of two or more extreme transient loads being applied simultaneously. Nominal Loads (Qn): The maximum actual loads expected in a building during its life (also called "service loads"). The magnitudes of loads, permanent and transient, computed using load models such as those found in ASCE 7. Nominal Strength (Rn): The capacity of a structure or member to resist effects of loads. The strength computed using theoretical models or by appropriate laboratory tests. Resistance Factor (φ): A factor that accounts for deviation of the actual strength from the nominal strength due to variations in member or material properties and uncertainties in the nominal load model. Strength Design: A method of proportioning structural members such that the effects of factored loads on the member do not exceed the design strength of the member: Σ(λQn) ≤ φRn. Also called "Load Resistance Factor Design" and "Ultimate Strength Design."

Live Load Reduction The effect on a building member of nominal occupancy live loads may often be reduced based on the loaded floor area supported by the member. A typical model used for computing reduced live load (as found in ASCE 7 and many building codes) is: Lreduced = Lnominal e0.25 + where: Lnominal AT KLLAT KLL

15 o $ 0.4Lnominal KLL AT

= nominal live load given in ASCE 7 or a building code = the cumulative floor tributary area supported by the member = area of influence supported by the member = ratio of area of influence to the tributary area supported by the member: KLL = 4 (typical columns) KLL = 2 (typical beams and girders)

Load Combinations using Strength Design (LRFD, USD) Nominal loads used in following combinations: D E L Lr R S W

= = = = = = =

dead loads earthquake loads live loads (floor) live loads (roof) rain load snow load wind load

Load factors λ: λD (dead load), λL (live load), etc. Basic combinations

Lr/S/R = largest of Lr, S, R L or 0.8W = larger of L, 0.8W

1.4D 1.2D + 1.6L + 0.5 (Lr /S/R) 1.2D + 1.6(Lr /S/R) + (L or 0.8W) 1.2D + 1.6W + L +0.5(Lr /S/R) 1.2D + 1.0E + L + 0.2S 0.9D + 1.6W 0.9D + 1.0E

CIVIL ENGINEERING

143

REINFORCED CONCRETE DESIGN ACI 318-05 US Customary units Definitions a = depth of equivalent rectangular stress block, in.

ASTM STANDARD REINFORCING BARS BAR SIZE

Ag = gross area of column, in2 As' = area of compression reinforcement, in2 Ast = total area of longitudinal reinforcement, in2 Av = area of shear reinforcement within a distance s, in. = width of compression face of member, in.

be = effective compression flange width, in. bw = web width, in. β1 = ratio of depth of rectangular stress block, a, to depth to neutral axis, c = 0.85 $ 0.85 - 0.05 d

f lc - 4, 000 n $ 0.65 1, 000

c

= distance from extreme compression fiber to neutral axis, in.

d

= distance from extreme compression fiber to centroid of nonprestressed tension reinforcement, in.

dt

= distance from extreme compression fiber to extreme tension steel, in.

Ec = modulus of elasticity = 33w1c.5

f lc , psi

εt = net tensile strain in extreme tension steel at nominal

AREA, IN

0.375 0.500 0.625 0.750 0.875 1.000 1.128 1.270 1.410 1.693 2.257

0.11 0.20 0.31 0.44 0.60 0.79 1.00 1.27 1.56 2.25 4.00

#3 #4 #5 #6 #7 #8 #9 #10 #11 #14 #18

As = area of tension reinforcement, in2

b

DIAMETER, IN.

2

WEIGHT, LB/FT

0.376 0.668 1.043 1.502 2.044 2.670 3.400 4.303 5.313 7.650 13.60

LOAD FACTORS FOR REQUIRED STRENGTH

U = 1.4 D U = 1.2 D + 1.6 L SELECTED ACI MOMENT COEFFICIENTS Approximate moments in continuous beams of three or more spans, provided: 1. 2. 3.

Span lengths approximately equal (length of longer adjacent span within 20% of shorter) Uniformly distributed load Live load not more than three times dead load

strength f lc = compressive strength of concrete, psi fy

= yield strength of steel reinforcement, psi

hf

= T-beam flange thickness, in.

Mc = factored column moment, including slenderness effect, in.-lb

Mu = coefficient * wu * Ln2 wu = factored load per unit beam length Ln = clear span for positive moment; average adjacent clear spans for negative moment

Column

+

Mn = nominal moment strength at section, in.-lb φMn= design moment strength at section, in.-lb −

Mu = factored moment at section, in.-lb

1 14

1 16

+

−

Pn = nominal axial load strength at given eccentricity, lb

1 10

Pu = factored axial force at section, lb

Spandrel

ρg = ratio of total reinforcement area to cross-sectional area of column = Ast /Ag s = spacing of shear ties measured along longitudinal axis of member, in. Vc = nominal shear strength provided by concrete, lb Vn = nominal shear strength at section, lb φVn = design shear strength at section, lb Vs = nominal shear strength provided by reinforcement, lb Vu = factored shear force at section, lb

beam

+ −

1 24

Unrestrained + end

1 14

+

−

1 10

−

1 11

−

1 11

−

1 11

−

1 10

−

−

1 11

1 16

1 11

+

End span CIVIL ENGINEERING

1 11

Ln

φPn = design axial load strength at given eccentricity, lb

144

−

1 16

1 11

1 16

Interior span

BEAMS − FLEXURE: φMn ≥ Mu

UNIFIED DESIGN PROVISIONS

Internal Forces and Strains d' Comp. strain Mu Cc Cs'

A' s

Pu

As

ε's

c d

dt

Ts Net tensile strain:

Strain Conditions 0.003 0.003

c

A' s

εt

0.003

c

For all beams Net tensile strain: a = β 1 c 0.003 ( dt − c ) 0.003 ( β1 dt − a ) εt = = c a Design moment strength: φMn where: φ = 0.9 [εt ≥ 0.005] φ = 0.48 + 83εt [0.004 ≤ εt < 0.005] Reinforcement limits: AS, max εt = 0.004 @ M n AS, min = larger

{

′ 3 f b d c w f y

or

200b d w f y

As,min limits need not be applied if As (provided) ≥ 1.33 As (required)

c

dt As

Singly-reinforced beams As,max = εt ≥ 0.005 Tensioncontrolled section: c ≤ 0.375 dt

0.005 > εt > 0.002

εt ≤ 0.002

Transition section

0.85 f c ' β1 b fy

3d t 7

As f y

Compressioncontrolled section:

a=

c ≥ 0.6 dt

Mn = 0.85 fc' a b (d −

0.85 f c′ b a a ) = As fy (d − ) 2 2

Doubly-reinforced beams Compression steel yields if:

Balanced Strain: εt = εy

0.003 As − As' ≥

A' s

dt

(

0.85 β1 f c′d ′ b fy

87, 000 87, 000 − f y

)

If compression steel yields:

As εt = εy =

( )

fy

As,max =

Es

Tension-controlled sections ( εt ≥ 0.005): φ = 0.9 Compression-controlled sections ( εt ≤ 0.002): φ = 0.70 Members with spiral reinforcement φ = 0.65 Members with tied reinforcement Transition sections (0.002 < εt < 0.005): Members with spiral reinforcement φ = 0.57 + 67εt Members with tied reinforcement φ = 0.48 + 83εt Shear and torsion φ = 0.75 φ = 0.65 Bearing on concrete

( 37d ) + A′ t

s

fy

( As − As′ ) f y

a = RESISTANCE FACTORS, φ

0.85 fc′β1 b

0.85 f c ' b

Mn = f y

[(

(

As − As′ ) d −

a 2

) + A′ (d − d ' )] s

If compression steel does not yield (two steps): 1. Solve for c: (87,000 − 0.85 f c ' ) As ' − As f y c c2 + 0.85 f c ' β1 b

(

)

−

87,000 As ' d ' =0 0.85 f c ' β1 b

CIVIL ENGINEERING

145

BEAMS − SHEAR: φV n ≥ V u

BEAMS − FLEXURE: φM n ≥ Mu (CONTINUED)

Beam width used in shear equations:

Doubly-reinforced beams (continued)

Compression steel does not yield (continued) 2. Compute Mn:

(

βc Mn = 0.85 b c β1 f c' d − 21

)

V n = Vc + V s

( )

Vc = 2 b w d f c ' Vs =

Effective flange width: 1/4 • span length b w + 16 • hf smallest

beam centerline spacing

Design moment strength: As f y a= 0.85 f c ' be

φVc

2

If a ≤ hf : As,max =

0.85 f c ' β1 be fy

( ) 3 dt 7

< Vu ≤ φ Vc

Smaller of: Av f y s= 50b w

a Mn = 0.85 f c' a b e (d − ) 2

Required spacing

s=

( )

0.85 f c 'β1 be 3 d t 7 fy

Redefine a: a =

+

As fy 0.85 f c ' bw

Mn = 0.85 f c' [hf (be − bw) (d −

V u > φ Vc

Vs =

Av f y 0.75 b w

If a > hf : As,max =

Av f y d

(may not exceed 8 bw d f c' ) s Required and maximum-permitted stirrup spacing, s φVc Vu ≤ : No stirrups required 2 φVc Vu > : Use the following table ( Av given): 2

T-beams − tension reinforcement in stem

=

bw (T-beams)

Nominal shear strength:

c − d' + As' c ( d − d ' ) 87,000

be

b (rectangular beams)

bw =

fc '

s=

Vu φ

− Vc

Av f y d Vs

0.85 f c ' (be − bw ) h f fy

− hf 2

Vs ≤ 4 bw d

h f (be − bw ) bw

) + a bw (d −

a )] 2

Maximum permitted spacing

Smaller of: d s= 2

fc '

Smaller of: d OR s= 2 s = 24"

OR s = 24"

Vs > 4 bw d

Smaller of: d s= 4 s = 12"

146

CIVIL ENGINEERING

fc '

SHORT COLUMNS Limits for main reinforcements: A ρg = st Ag 0.01 ≤ ρ g ≤ 0.08 Definition of a short column: 12 M 1 KL ≤ 34 − M2 r KL = Lcol

where:

clear height of column [assume K = 1.0]

r = 0.288h rectangular column, h is side length perpendicular to buckling axis ( i.e., side length in the plane of buckling) r = 0.25h circular column, h = diameter

Concentrically-loaded short columns: φPn ≥ Pu M1 = M2 = 0 KL ≤ 22 r Design column strength, spiral columns: φ = 0.70 φPn = 0.85φ [0.85 fc ' (A g − Ast ) + A st fy] Design column strength, tied columns: φ = 0.65 φPn = 0.80φ [0.85 fc' ( A g − Ast ) + A st fy] Short columns with end moments: Mu = M 2 or Mu = P u e Use Load-moment strength interaction diagram to: 1. Obtain φPn at applied moment Mu 2. Obtain φPn at eccentricity e 3. Select As for Pu, M u

M 1 = smaller end moment M 2 = larger end moment M1 M2

positive if M1, M2 cause single curvature negative if M1, M2 cause reverse curvature

LONG COLUMNS − Braced (non-sway) frames Definition of a long column: 12 M 1 KL > 34 − r M2 Critical load: Pc =

π2 E I = ( KL ) 2

where:

π2 E I ( Lcol ) 2

Long columns with end moments: M 1 = smaller end moment M 2 = larger end moment M1 positive if M 1 , M 2 produce single curvature M2 C m = 0.6 +

EI = 0.25 Ec Ig Mc =

Concentrically-loaded long columns: emin = ( 0.6 + 0.03h) minimum eccentricity M 1 = M 2 = Pu emin (positive curvature) KL > 22 r M2 Mc = Pu 1− 0.75 Pc

0.4 M 1 ≥ 0.4 M2

Cm M 2 ≥ M2 Pu 1− 0.75 Pc

Use Load-moment strength interaction diagram to design/analyze column for Pu , M u

Use Load-moment strength interaction diagram to design/analyze column for Pu , M u

CIVIL ENGINEERING

147

Pn Pu = f lc Ag zf lc Ag Kn =

Rn =

Pne Pue = f lc Agh zf lc Agh

GRAPH A.11 Column strength interaction diagram for rectangular section with bars on end faces and γ = 0.80 (for instructional use only). Nilson, Arthur H., David Darwin, and Charles W. Dolan, Design of Concrete Structures, 13th ed., McGraw-Hill, New York, 2004.

148

CIVIL ENGINEERING

Pn Pu = f lc Ag zf lc Ag Kn =

Rn =

Pne Pue = l f c Agh zf lc Agh

GRAPH A.15 Column strength interaction diagram for circular section γ = 0.80 (for instructional use only). Nilson, Arthur H., David Darwin, and Charles W. Dolan, Design of Concrete Structures, 13th ed., McGraw-Hill, New York, 2004.

CIVIL ENGINEERING

149

STEEL STRUCTURES (AISC Manual, 13th Edition) LRFD, E = 29,000 ksi

Shear – unstiffened beams

Definitions (AISC Specifications): LRFD Available strength: Product of the nominal strength and a resistance factor (same as "design strength" by ASCE 7)

Rolled W-shapes for Fy ≤ 50 ksi: φ = 1.0

Required strength: The controlling combination of the nominal loads multiplied by load factors (same as critical factored load combination by ASCE 7)

Built-up I-shaped beams for Fy $ 50 ksi: φ = 0.90

BEAMS Beam flexure strength of rolled compact sections, flexure about x-axis, Fy = 50 ksi

418 < h # 522 : Fy tw Fy

Compact section criteria: bf Flange: # m pf = 64.7 2t f Fy

For rolled sections, use tabulated values of

= = = = =

Cb =

h 418 tw # Fy :

zVn = z _0.6Fy i Aw

zVn = z _0.6Fy i Aw >

zVn = z _0.6Fy i Aw > bf h , 2t f tw

Effect of lateral support on available moment capacity: Lb φ Lp, Lr φMp φMr

φVn = φ(0.6Fy)Aw

h 522 tw > Fy :

h 640.3 tw # m pw = Fy

Web:

Aw = dtw

unbraced length of beam segment resistance factor for bending = 0.90 length limits φFyZx AISC Table J3-2 φ0.7FxSx

12.5 Mmax # 3.0 2.5Mmax + 3MA + 4MB + 3MC

Lb # Lp: zMn = zMp

= Cb 8zMp - BF _ Lb - LpiB # zMp

Lb > Lr: R V 2 S r 2E Lb W J d n WSx # zMp zMn = zCb S 2 1 + 0.078 c S h m r ts z o S d Lb n W S rtx W T4 X IyCw where: rts = Sx2 ho = distance between flange centroids = d – tf J, Cw

= torsion constants ( AISC Table 1-1 )

See AISC Table 3-10 for φMn vs. Lb curves

150

CIVIL ENGINEERING

220, 000 2 H _h tw i Fy

COLUMNS Column effective length KL AISC Table C-C2.2 – Approximate Values of Effective Length Factor, K AISC Figures C-C2.3 and C-C2.4 – Alignment Charts Column capacity – available strength φ = 0.9 φPn = φFcrA Slenderness ratio: KL r

Lp< Lb # Lr: Lb - Lp oH zMn = Cb >zMp - _zMp - zMr i e Lr - Lp

418 H _h tw i Fy

KL # 802.1: r Fy KL > 802.1: r Fy

2 Fy b KL l r

= 286, 220 G zFcr = 0.9 0.658 Fy zFcr =

225, 910 2 b KL l r

AISC Table 4-22:

Available Critical Stress (φFcr) for Compression Members

AISC Table 4-1:

Available Force (φPn) in Axial Compression, kips (Fy = 50 ksi)

BEAM-COLUMNS Sidesway-prevented, x-axis bending, transverse loading between supports (no moments at ends), ends pinned permitting rotation in plane of bending

TRANSVERSE LOADS

Pnt

TENSION MEMBERS Flat bars or angles, bolted or welded Definitions: Bolt diameter: db Hole diameter: dh = db + 1/16"

FOR NO SIDESWAY: Mlt = 0 Plt = 0

L

Gross width of member: bg Member thickness: t Gross area: Ag = bg t (use tabulated areas for angles) Net area (parallel holes ): An = (bg – Σdh) t Net area (staggered holes):

Mnt DIAGRAM

Pnt

An = `bg - Rdh + s 2 4g j t

s = longitudinal spacing of consecutive holes Required strengths:

Pr = Pnt Mr = B1 Mnt

where: B1 =

Cm

P 1- r Pel Cm = 1.0 for conditions stated above

g = transverse spacing between lines of holes Effective area (bolted members): Ae = UAn

U = 1.0 ^flat barsh U = 1 - x/L _anglesi

Effective area (welded members):

2 Pel = r EI2 with respect to bending axis ] KLgx

Flat bars or angles with transverse welds: U = 1.0 Flat bars of width "w", longitudinal welds of length "L" only: U = 1.0 (L ≥ 2w) U = 0.87 (2w > L ≥ 1.5w) U = 0.75 (1.5w > L > w)

Strength limit state: Pr $ 0.2: zPn

Pr Mr # 1.0 + 89 zPn zMnx

Pr < 0.2: zPn

Pr Mr # 1.0 + 2 _zPni zMnx

Ae = UAn

Angles with longitudinal welds only U = 1 - x/L

where: zPn = compression strength with respect to weak axis (y -axis) zMn = bending strength with respect to bending axis (x -axis)

Limit states and available strengths: Yielding:

φy = 0.90 φTn = φy Fy Ag

Fracture:

φf = 0.75 φTn = φf Fu Ae

Block shear:

φ Ubs Agv Anv Ant

= = = = =

zTn =

0.75 1.0 (flat bars and angles) gross area for shear net area for shear net area for tension

smaller

0.75 Fu 70.6Anv + Ubs Ant A 0.75 80.6FyAgv + UbsFuAnt B

CIVIL ENGINEERING

151

BOLT STRENGTHS Definitions: db = bolt diameter Le = end distance between center of bolt hole and back edge of member (in direction of applied force) s = spacing between centers of bolt holes (in direction of applied force) Bolt tension and shear available strengths (kips/bolt): Standard size holes. Shear strengths are single shear. Slip-critical connections are Class A faying surface.

Bearing strength of connected member at bolt hole: The bearing resistance of the connection shall be taken as the sum of the bearing resistances of the individual bolts. Available strength (kips/bolt/inch thickness): zrn = z1.2Lc Fu # z2.4db Fu z = 0.75 Lc = clear distance between edge of hole and edge of adjacent hole, or edge of member, in direction of force

Available bolt strength, φrn BOLT SIZE, in. 3/4 7/8 1 TENSION, kips A 325 29.8 40.6 53.0 A 307 10.4 20.3 26.5 SHEAR (BEARING-TYPE CONNECTION), kips A 325-N 15.9 21.6 28.3 A 325-X 19.9 27.1 35.3 A 306 7.95 10.8 14.1 SHEAR (SLIP-CRITICAL CONNECTION, SLIP IS A SERVICEABILITY LIMIT STATE), kips A 325-SC 11.1 15.4 20.2 SHEAR (SLIP-CRITICAL CONNECTION, SLIP IS A STRENGTH LIMIT STATE), kips A 325-SC 9.41 13.1 17.1

Lc = s - Dh

BOLT DESIGNATION

For slip-critical connections: Slip is a serviceability limit state if load reversals on bolted connection could cause localized failure of the connection (such as by fatigue). Slip is a strength limit state if joint distortion due to slip causes either structure instability or a significant increase of external forces on structure.

interior holes

D Lc = Le - h 2

end hole

Db = bolt diameter Dh = hole diameter = bolt diameter + clearance s

= center -to -center spacing of interior holes

Le = end distance (center of end hole to end edge of member) Available bearing strength, φrn, at bolt holes, kips/bolt/in. thickness * FU , ksi CONNECTED MEMBER

BOLT SIZE, in. 3/4 s = 2

2

7/8

1

/ 3 d b (MINIMUM PERMITTED)

58 65

62.0 69.5

58 65

78.3 87.7

72.9 81.7

83.7 93.8

91.3 102

101 113

40.8 45.7

37.5 42.0

79.9 89.6

76.7 85.9

s = 3"

Le = 1 58 65

44.0 49.4

58 65

78.3 87.7

1

/4 "

L e = 2"

* Dh = Db + 1/16 "

152

CIVIL ENGINEERING

Y

tf

Table 1-1: W-Shapes Dimensions and Properties X tw

X

d

bf

Area Depth Web Shape

A 2

d

In.

bf In.

tf

Compact

rts

ho

section

Tors. Prop. J in.

Cw 4

6

Axis X-X I

S 4

3

r

Z

I 3

r 4

in.

23.7 0.415

8.97

In. bf/2tf h/tw 0.585 7.66 52.0

in.

20.1

2.30

23.1

1.87

9430

1830

154 9.55

177

70.4

1.87

W24X62

18.2

23.7 0.430

7.04

0.590 5.97

49.7

1.75

23.2

1.71

4620

1550

131 9.23

153

34.5

1.38

W24X55

16.3

23.6 0.395

7.01

0.505 6.94

54.1

1.71

23.1

1.18

3870

1350

114 9.11

134

29.1

1.34

W21X73

21.5

21.2 0.455

8.30

0.740 5.60

41.2

2.19

20.5

3.02

7410

1600

151 8.64

172

70.6

1.81

W21X68

20.0

21.1 0.430

8.27

0.685 6.04

43.6

2.17

20.4

2.45

6760

1480

140 8.60

160

64.7

1.80

W21X62

18.3

21.0 0.400

8.24

0.615 6.70

46.9

2.15

20.4

1.83

5960

1330

127 8.54

144

57.5

1.77

W21X55

16.2

20.8 0.375

8.22

0.522 7.87

50.0

2.11

20.3

1.24

4980

1140

110 8.40

126

48.4

1.73

W21X57

16.7

21.1 0.405

6.56

0.650 5.04

46.3

1.68

20.4

1.77

3190

1170

111 8.36

129

30.6

1.35

W21X50

14.7

20.8 0.380

6.53

0.535 6.10

49.4

1.64

20.3

1.14

2570

984

94.5 8.18

110

24.9

1.30

W21X48

14.1

20.6 0.350

8.14

0.430 9.47

53.6

2.05

20.2 0.803

3950

959

93.0 8.24

107

38.7

1.66

W21X44

13.0

20.7 0.350

6.50

0.450 7.22

53.6

1.60

20.2 0.770

2110

843

81.6 8.06

95.4

20.7

1.26

W18X71

20.8

18.5 0.495

7.64

0.810 4.71

32.4

2.05

17.7

4700

1170

127 7.50

146

60.3

1.70

W18X65

19.1

18.4 0.450

7.59

0.750 5.06

35.7

2.03

17.6

2.73

4240

1070

117 7.49

133

54.8

1.69

W18X60

17.6

18.2 0.415

7.56

0.695 5.44

38.7

2.02

17.5

2.17

3850

984

108 7.47

123

50.1

1.68

W18X55

16.2

18.1 0.390

7.53

0.630 5.98

41.1

2.00

17.5

1.66

3430

890

98.3 7.41

112

44.9

1.67

W18X50

14.7

18.0 0.355

7.50

0.570 6.57

45.2

1.98

17.4

1.24

3040

800

88.9 7.38

101

40.1

1.65

W18X46

13.5

18.1 0.360

6.06

0.605 5.01

44.6

1.58

17.5

1.22

1720

712

78.8 7.25

90.7

22.5

1.29

W18X40

11.8

17.9 0.315

6.02

0.525 5.73

50.9

1.56

17.4 0.810

1440

612

68.4 7.21

78.4

19.1

1.27

W16X67

19.7

16.3 0.395

10.2

0.67 7.70

35.9

2.82

15.7

2.39

7300

954

117 6.96

130

119

2.46

W16X57

16.8

16.4 0.430

7.12

0.715 4.98

33.0

1.92

15.7

2.22

2660

758

92.2 6.72

105

43.1

1.60

W16X50

14.7

16.3 0.380

7.07

0.630 5.61

37.4

1.89

15.6

1.52

2270

659

81.0 6.68

92.0

37.2

1.59

W16X45

13.3

16.1 0.345

7.04

0.565 6.23

41.1

1.88

15.6

1.11

1990

586

72.7 6.65

82.3

32.8

1.57

W16X40

11.8

16.0 0.305

7.00

0.505 6.93

46.5

1.86

15.5

0.794

1730

518

64.7 6.63

73.0

28.9

1.57

W16X36

10.6

15.9 0.295

6.99

0.430 8.12

48.1

1.83

15.4

0.545

1460

448

56.5 6.51

64.0

24.5

1.52

W14X74

21.8

14.2 0.450

10.1

0.785 6.41

25.4

2.82

13.4

3.87

5990

795

112 6.04

126

134

2.48

W14X68

20.0

14.0 0.415

10.0

0.720 6.97

27.5

2.80

1303

3.01

5380

722

103 6.01

115

121

2.46

W14X61

17.9

13.9 0.375

9.99

0.645 7.75

30.4

2.78

13.2

2.19

4710

640

92.1 5.98

102

107

2.45

W14X53

15.6

13.9 0.370

8.06

0.660 6.11

30.9

2.22

13.3

1.94

2540

541

77.8 5.89

87.1

57.7

1.92

W14X48

14.1

13.8 0.340

8.03

0.595 6.75

33.6

2.20

13.2

1.45

2240

484

70.2 5.85

78.4

51.4

1.91

W12X79

23.2

12.4 0.470

12.1

0.735 8.22

20.7

3.43

11.6

3.84

7330

662

107 5.34

119

216

3.05

W12X72

21.1

12.3 0.430

12.0

0.670 8.99

22.6

3.40

11.6

2.93

6540

597

97.4 5.31

108

195

3.04

W12X65

19.1

12.1 0.390

12.0

0.605 9.92

24.9

3.38

11.5

2.18

5780

533

87.9 5.28

96.8

174

3.02

W12X58

17.0

12.2 0.360

10.0

0.640 7.82

27.0

2.82

11.6

2.10

3570

475

78.0 5.28

86.4

107

2.51

W12X53

15.6

12.1 0.345

9.99

0.575 8.69

28.1

2.79

11.5

1.58

3160

425

70.6 5.23

77.9

95.8

2.48

W12X50

14.6

12.2 0.370

8.08

0.640 6.31

26.8

2.25

11.6

1.71

1880

391

64.2 5.18

71.9

56.3

1.96

W12X45

13.1

12.1 0.335

8.05

0.575 7.00

29.6

2.23

11.5

1.26

1650

348

57.7 5.15

64.2

50.0

1.95

W12X40

11.7

11.9 0.295

8.01

0.515 7.77

33.6

2.21

11.4

0.906

1440

307

51.5 5.13

57.0

44.1

1.94

3.49

in

In.

In.

In.

Axis Y-Y

W24X68

In.

In.

tw

Flange

In.

In.

In.

W10x60

17.6

10.2 0.420

10.1

0.680 7.41

18.7

2.88

9.54

2.48

2640

341

66.7 4.39

74.6

116

2.57

W10x54

15.8

10.1 0.370

10.0

0.615 8.15

21.2

2.86

9.48

1.82

2320

303

60.0 4.37

66.6

103

2.56

W10x49

14.4

10.0 0.340

10.0

0.560 8.93

23.1

2.84

9.42

1.39

2070

272

54.6 4.35

60.4

93.4

2.54

W10x45

13.3

10.1 0.350

8.02

0.620 6.47

22.5

2.27

9.48

1.51

1200

248

49.1 4.32

54.9

53.4

2.01

W10x39

11.5

9.92 0.315

7.99

0.530 7.53

25.0

2.24

9.39

0.976

992

209

42.1 4.27

46.8

45.0

1.98

Adapted from Steel Construction Manual, 13th ed., AISC, 2005.

CIVIL ENGINEERING

153

AISC Table 3-2

Zx

φb = 0.90 Ix φvVnx in.4 kips

Shape

Zx in.3

φbMpx kip-ft

φbMrx kip-ft

BF kips

Lp ft.

Lr ft.

W24 x 55 W18 x 65 W12 x 87 W16 x 67 W10 x 100 W21 x 57

134 133 132 130 130 129

503 499 495 488 488 484

299 307 310 307 294 291

22.2 14.9 5.76 10.4 4.01 20.1

4.73 5.97 10.8 8.69 9.36 4.77

13.9 18.8 43.0 26.1 57.7 14.3

1350 1070 740 954 623 1170

251 248 194 194 226 256

W21 x 55 W14 x 74 W18 x 60 W12 x 79 W14 x 68 W10 x 88

126 126 123 119 115 113

473 473 461 446 431 424

289 294 284 281 270 259

16.3 8.03 14.5 5.67 7.81 3.95

6.11 8.76 5.93 10.8 8.69 9.29

17.4 31.0 18.2 39.9 29.3 51.1

1140 795 984 662 722 534

234 191 227 175 175 197

W18 x 55

112

420

258

13.9

5.90

17.5

890

212

W21 x 50 W12 x 72

110 108

413 405

248 256

18.3 5.59

4.59 10.7

13.6 37.4

984 597

237 158

W21 x 48 W16 x 57 W14 x 61 W18 x 50 W10 x 77 W12 x 65

107 105 102 101 97.6 96.8

398 394 383 379 366 356

244 242 242 233 225 231

14.7 12.0 7.46 13.1 3.90 5.41

6.09 5.56 8.65 5.83 9.18 11.9

16.6 18.3 27.5 17.0 45.2 35.1

959 758 640 800 455 533

217 212 156 192 169 142

W21 x 44 W16 x 50 W18 x 46 W14 x 53 W12 x 58 W10 x 68 W16 x 45

95.4 92.0 90.7 87.1 86.4 85.3 82.3

358 345 340 327 324 320 309

214 213 207 204 205 199 191

16.8 11.4 14.6 7.93 5.66 3.86 10.8

4.45 5.62 4.56 6.78 8.87 9.15 5.55

13.0 17.2 13.7 22.2 29.9 40.6 16.5

843 659 712 541 475 394 586

217 185 195 155 132 147 167

W18 x 40 W14 x 48 W12 x 53 W10 x 60

78.4 78.4 77.9 74.6

294 294 292 280

180 184 185 175

13.3 7.66 5.48 3.80

4.49 6.75 8.76 9.08

13.1 21.1 28.2 36.6

612 484 425 341

169 141 125 129

W16 x 40 W12 x 50 W8 x 67 W14 x 43 W10 x 54

73.0 71.9 70.1 69.6 66.6

274 270 263 261 250

170 169 159 164 158

10.1 5.97 2.60 7.24 3.74

5.55 6.92 7.49 6.68 9.04

15.9 23.9 47.7 20.0 33.7

518 391 272 428 303

146 135 154 125 112

W18 x 35 W12 x 45 W16 x 36 W14 x 38 W10 x 49 W8 x 58 W12 x 40 W10 x 45

66.5 64.2 64.0 61.5 60.4 59.8 57.0 54.9

249 241 240 231 227 224 214 206

151 151 148 143 143 137 135 129

12.1 5.75 9.31 8.10 3.67 2.56 5.50 3.89

4.31 6.89 5.37 5.47 8.97 7.42 6.85 7.10

12.4 22.4 15.2 16.2 31.6 41.7 21.1 26.9

510 348 448 385 272 228 307 248

159 121 140 131 102 134 106 106

φbMrx = φb(0.7Fy)Sx

BF =

φbMpx − φbMrx


154

Fy = 50 ksi

W Shapes – Selection by Zx

CIVIL ENGINEERING

Lr − Lp

Fy = 50 ksi φ = 0.90

φ Mn kip-ft LRFD

AVAILABLE MOMENT VS. UNBRACED LENGTH

W1 6X

X62 W24

8 X6

67

W1

X57

5 1X5 W2

W2 1

7

X7

14 4

X55

8

X7

9

W21

8

X7

14

W

X68

1

4

8

12

X62

67 6X 73 W21X

68

2

W21

5

X7

W1

W24X

X6

W

W1

0X

77

X55

8X 50

W

6 W1

77

W

12

X6

5

68

W21X

5

16

8X6

W1 1

14

0 8X6 W1 5 1X5 W2 62 W24X

55

8X

X57

W1

W21 8 X5 12 W 0 5 8X

W1

53 X50 14X W 50

12

0X

X6

8

W21 6X W1

4

1X4 W2

46

8X

W1

10

W1

14

7 X5

1X4

W2 W10X68

8

X8

14

8X6

X62

W1

W24

55

W24

8X

W1 W1

X50

W21

330

300

10

W

5

60

57

8X

1X5

W1

8X7

1

W2

X6

14

6X

345

W

W1

W

W1

Available Moment φ Mn ( foot-kips)

W

83 W21X

6 W24X7

77

6X

1 8X7 W1

62

W1

1X W2

W24

W

12

W12X65

6

00

12

X57

1X4

W21

W2

375

X1

W

X8

55 8X W1

390

315

10

W12X72

405

360

W

5

W14X68

2 X8 14 W X68 W24 X73 W21

W12X79

W10X88

420

X68 W21

450

435

TABLE 3-10 (CONTINUED) W SHAPES

18

20

22

UNBRACED LENGTH (0.5-ft INCREMENTS) From Steel Construction Manual, 13th ed., AISC, 2005.

CIVIL ENGINEERING

155

TABLE C-C2.2 APPROXIMATE VALUES OF EFFECTIVE LENGTH FACTOR, K BUCKLED SHAPE OF COLUMN IS SHOWN BY DASHED LINE.

(a)

(b)

THEORETICAL K VALUE RECOMMENDED DESIGN VALUE WHEN IDEAL CONDITIONS ARE APPROXIMATED

0.5

0.7

1.0

1.0

2.0

2.0

0.65

0.80

1.2

1.0

2.10

2.0

END CONDITION CODE

(c)

(d)

(e)

(f)

ROTATION FIXED AND TRANSLATION FIXED ROTATION FREE AND TRANSLATION FIXED ROTATION FIXED AND TRANSLATION FREE ROTATION FREE AND TRANSLATION FREE

FOR COLUMN ENDS SUPPORTED BY, BUT NOT RIGIDLY CONNECTED TO, A FOOTING OR FOUNDATION, G IS THEORETICALLY INFINITY BUT UNLESS DESIGNED AS A TRUE FRICTION-FREE PIN, MAY BE TAKEN AS 10 FOR PRACTICAL DESIGNS. IF THE COLUMN END IS RIGIDLY ATTACHED TO A PROPERLY DESIGNED FOOTING, G MAY BE TAKEN AS 1.0. SMALLER VALUES MAY BE USED IF JUSTIFIED BY ANALYSIS.

AISC Figure C-C2.3 Alignment chart, sidesway prevented GA 50.0 10.0 5.0 4.0 3.0

K

AISC Figure C-C2.4 Alignment chart, sidesway not prevented GB

1.0

0.9

50.0 10.0 5.0 4.0 3.0

GA 100.0 50.0 30.0 20.0

10.0 0.8

0.7

1.0 0.9 0.8 0.7 0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

20.0 10.0 5.0 4.0

0.6

3.0

8.0 7.0 6.0 5.0 4.0

2.0

3.0

CIVIL ENGINEERING

4.0 3.0

2.0

2.0 1.5

0.2 1.0

0.1

0.5

From Steel Construction Manual, 13th ed., AISC, 2005.

156

10.0 8.0 7.0 6.0 5.0

1.0 0.1

0.0

100.0 50.0 30.0 20.0

2.0

2.0

1.0 0.9 0.8 0.7 0.6

GB

K

0.0

0.0

1.0

0.0

AISC Table 4-22 Available Critical Stress φcFcr for Compression Members Fy = 50 ksi φc = 0.90 KL r 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

φFcr ksi 45.0 45.0 45.0 44.9 44.9 44.9 44.8 44.8 44.7 44.7 44.6 44.5 44.4 44.4 44.3 44.2 44.1 43.9 43.8 43.7 43.6 43.4 43.3 43.1 43.0 42.8 42.7 42.5 42.3 42.1 41.9 41.8 41.6 41.4 41.2 40.9 40.7 40.5 40.3 40.0

KL r 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

φFcr ksi 39.8 39.5 39.3 39.1 38.8 38.5 38.3 38.0 37.7 37.5 37.2 36.9 36.7 36.4 36.1 35.8 35.5 35.2 34.9 34.6 34.3 34.0 33.7 33.4 33.0 32.7 32.4 32.1 31.8 31.4 31.1 30.8 30.5 30.2 29.8 29.5 29.2 28.8 28.5 28.2

KL r 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120

φFcr ksi 27.9 27.5 27.2 26.9 26.5 26.2 25.9 25.5 25.2 24.9 24.6 24.2 23.9 23.6 23.3 22.9 22.6 22.3 22.0 21.7 21.3 21.0 20.7 20.4 20.1 19.8 19.5 19.2 18.9 18.6 18.3 18.0 17.7 17.4 17.1 16.8 16.5 16.2 16.0 15.7

KL r 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160

φFcr ksi 15.4 15.2 14.9 14.7 14.5 14.2 14.0 13.8 13.6 13.4 13.2 13.0 12.8 12.6 12.4 12.2 12.0 11.9 11.7 11.5 11.4 11.2 11.0 10.9 10.7 10.6 10.5 10.3 10.2 10.0 9.91 9.78 9.65 9.53 9.40 9.28 9.17 9.05 8.94 8.82

KL r 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200

φFcr ksi 8.72 8.61 8.50 8.40 8.30 8.20 8.10 8.00 7.89 7.82 7.73 7.64 7.55 7.46 7.38 7.29 7.21 7.13 7.05 6.97 6.90 6.82 6.75 6.67 6.60 6.53 6.46 6.39 6.32 6.26 6.19 6.13 6.06 6.00 5.94 5.88 5.82 5.76 5.70 5.65


CIVIL ENGINEERING

157

158

CIVIL ENGINEERING

Effective length KL (ft) with respect to least radius of gyration ry

670 639 608 576 544 512 480 448 387 329 281 242 211 185 164 146 131 119

734

701

667

632

598

563

528

494

428

365

311

268

234

205

182

162

146

131

13

14

15

16

17

18

19

20

22

24

26

28

30

32

34

36

38

40

755

826

10

700

781

853

9

766

804

878

8

12

826

901

7

728

844

922

6

797

899

980

0

11

68

74

105

117

130

146

165

187

215

250

293

345

400

428

457

486

515

543

572

599

626

652

677

700

721

740

757

806

61

W14

Heavy line indicates KL/r equal to or greater than 200

Shape wt/ft

Selected W14, W12, W10

88.1

100

115

133

157

186

226

250

278

308

338

369

401

433

465

497

528

557

585

610

633

702

53

89.9

103

120

140

167

202

224

250

276

304

332

361

391

420

449

477

504

529

552

573

636

48

105

117

130

146

165

187

215

249

293

342

393

420

446

473

500

527

553

578

603

627

649

670

689

707

722

767

58

93.9

104

116

130

147

167

192

222

261

306

353

378

402

427

452

477

501

525

547

569

590

610

628

644

659

701

53

82.9

97.7

112

130

153

182

220

244

270

297

326

354

384

413

443

471

499

526

551

574

595

657

50

W12

76.4

87.0

99.8

116

136

162

196

217

241

265

291

317

343

370

396

422

448

472

495

516

534

590

45

114

127

141

158

179

203

233

270

317

367

420

447

474

501

528

555

581

606

631

655

677

698

717

734

750

794

60

102

113

126

141

159

181

208

241

282

327

375

399

423

448

472

496

520

543

565

586

607

625

643

658

672

712

54

91.3

101

113

126

143

162

186

216

254

295

338

360

383

405

428

450

471

493

513

533

551

569

585

599

612

649

49

W10

82.1

93.4

107

124

146

174

210

233

256

281

306

332

358

384

410

435

460

483

505

525

543

597

45

Fy = 50 ksi φc = 0.90


67.3

76.6

88.0

102

120

142

172

191

212

234

257

280

304

328

351

375

397

419

439

458

475

526

40

AISC Table 4–1 Available Strength in Axial Compression, kips—W shapes LRFD: φPn

68.9

78.4

90.0

104

122

146

176

195

216

238

260

282

305

328

351

373

395

415

435

452

469

516

39

ENVIRONMENTAL ENGINEERING For information about environmental engineering refer to the ENVIRONMENTAL ENGINEERING section. HYDROLOGY

Storativity or storage coefficient, S, of an aquifer is the volume of water taken into or released from storage per unit surface area per unit change in potentiometric (piezometric) head.

NRCS (SCS) Rainfall-Runoff ^ h Q = P - 0.2S P + 0.8S 1, 000 S= 10 CN 1, 000 CN = S + 10

WELL DRAWDOWN:

2

Q GROUNDWATER LEVEL

P

= precipitation (inches),

S

= maximum basin retention (inches),

r2

r1

Q = runoff (inches), and CN = curve number.

PERMEABLE SOIL

h2

Rational Formula Q = CIA, where A

= watershed area (acres),

C

= runoff coefficient,

I

= rainfall intensity (in./hr), and

h1 BOTTOM OF AQUIFER

Q = peak discharge (cfs).

Dupuit's formula

Darcy’s law

Q=

Q = –KA(dh/dx), where

rk ` h22 - h12j r ln d r2 n 1

Q = discharge rate (ft3/sec or m3/s),

where

K

= hydraulic conductivity (ft/sec or m/s),

Q = flow rate of water drawn from well (cfs)

h

= hydraulic head (ft or m), and

k 2

2

A

= cross-sectional area of flow (ft or m ).

q

= –K(dh/dx) q = specific discharge or Darcy velocity

v

= q/n = –K/n(dh/dx) v = average seepage velocity n = effective porosity

Unit hydrograph: The direct runoff hydrograph that would result from one unit of effective rainfall occurring uniformly in space and time over a unit period of time.

= permeability of soil (fps)

h1 = height of water surface above bottom of aquifer at perimeter of well (ft) h2 = height of water surface above bottom of aquifer at distance r2 from well centerline (ft) r1 = radius to water surface at perimeter of well, i.e., radius of well (ft) r2 = radius to water surface whose height is h2 above bottom of aquifer (ft) ln = natural logarithm

Transmissivity, T, is the product of hydraulic conductivity and thickness, b, of the aquifer (L2T –1).

CIVIL ENGINEERING

159

Ratio of Minimum or of Peak-to-Average Daily Sewage Flow

SEWAGE FLOW RATIO CURVES

0.2

Curve A2: P 5 Curve B: 14 +1 4+ P P 18 Curve G: + 4+ P

Population in Thousands (P)

HYDRAULIC-ELEMENTS GRAPH FOR CIRCULAR SEWERS

Design and Construction of Sanitary and Storm Sewers, Water Pollution Control Federation and American Society of Civil Engineers, 1970.

160

CIVIL ENGINEERING

Open-Channel Flow Specific Energy

Manning’s Equation 2

aQ E = aV + y = + y, where 2g 2gA 2 2

E Q V y A α

= = = = = =

specific energy, discharge, velocity, depth of flow, cross-sectional area of flow, and kinetic energy correction factor, usually 1.0.

Critical Depth = that depth in a channel at minimum specific energy Q2 A3 = g T where Q and A are as defined above, g = acceleration due to gravity, and T = width of the water surface. For rectangular channels 1 3

q2 yc = e g o , where yc q B g

= = = =

critical depth, unit discharge = Q/B, channel width, and acceleration due to gravity.

Froude Number = ratio of inertial forces to gravity forces Fr = V , where gyh V = yh =

velocity, and hydraulic depth = A/T

Specific Energy Diagram

y

2 3 1 Q= K n AR S

2

Q = discharge (ft3/sec or m3/s), K = 1.486 for USCS units, 1.0 for SI units, A = cross-sectional area of flow (ft2 or m2), R = hydraulic radius = A/P (ft or m), P = wetted perimeter (ft or m), S = slope of hydraulic surface (ft/ft or m/m), and n = Manning’s roughness coefficient. Normal depth (uniform flow depth) AR 2 3 =

Qn KS1 2

Weir Formulas Unsubmerged suppressed Q = CLH 3/2 Unsubmerged contracted Q = C(L − 0.2H)H 3/2 V-Notch Q = CH 5/2, where Q = discharge (cfs or m3/s), C = 3.33 for submerged rectangular weir (USCS units), C = 1.84 for submerged rectangular weir (SI units), C = 2.54 for 90° V-notch weir (USCS units), C = 1.40 for 90° V-notch weir (SI units), L = weir length (ft or m), and H = head (depth of discharge over weir) ft or m. Hazen-Williams Equation V = k1CR0.63S0.54, where C = roughness coefficient, k1 = 0.849 for SI units, and k1 = 1.318 for USCS units, R = hydraulic radius (ft or m), S = slope of energy grade line, = hf /L (ft/ft or m/m), and V = velocity (ft/sec or m/s).

1 1

Values of Hazen-Williams Coefficient C Pipe Material

E=

αV 2 +y 2g

Alternate depths: depths with the same specific energy. Uniform flow: a flow condition where depth and velocity do not change along a channel.

Concrete (regardless of age) Cast iron: New 5 yr old 20 yr old Welded steel, new Wood stave (regardless of age) Vitrified clay Riveted steel, new Brick sewers Asbestos-cement Plastic

C 130 130 120 100 120 120 110 110 100 140 150 CIVIL ENGINEERING

161

For additional fluids information, see the FLUID MECHANICS section.

TRANSPORTATION U.S. Customary Units a = deceleration rate (ft/sec2) A = absolute value of algebraic difference in grades (%) e = superelevation (%) f = side friction factor ± G = percent grade divided by 100 (uphill grade "+") h1 = height of driver’s eyes above the roadway surface (ft) h2 = height of object above the roadway surface (ft) L = length of curve (ft) Ls = spiral transition length (ft) R = radius of curve (ft) S = stopping sight distance (ft) t = driver reaction time (sec) V = design speed (mph) v = vehicle approach speed (fps) W = width of intersection, curb-to-curb (ft) l = length of vehicle (ft) y = length of yellow interval to nearest 0.1 sec (sec) r = length of red clearance interval to nearest 0.1 sec (sec) Vehicle Signal Change Interval v y=t+ 2a ! 64.4 G r = W v+ l Stopping Sight Distance S = 1.47Vt +

162

V2 30 cb a l ! G m 32.2

CIVIL ENGINEERING

Transportation Models See INDUSTRIAL ENGINEERING for optimization models and methods, including queueing theory.

DENSITY k (veh/mi)

CAPACITY

CAPACITY

SPEED v (mph)

SPEED v (mph)

VOLUME q (veh/hr)

Traffic Flow Relationships (q = kv)

VOLUME q (veh/hr)

DENSITY k (veh/mi)

Vertical Curves: Sight Distance Related to Curve Length S ≤ L Crest Vertical Curve

AS2

L =

General equation:

Standard Criteria:

h 1 = 3.50 ft and h2 = 2.0 ft:

L =

L = 2S −

AS2 2,158

L = 2S −

L =

AS2 400 + 3.5 S L =

(based on riding comfort) L =

(based on adequate sight distance under an overhead structure to see an object beyond a sag vertical curve)

AS2 h +h 800 C − 1 2 2

(

(

h1 + h2

)

2

A

2,158 A

L = 2S −

Sag Vertical Curve

Sag Vertical Curve

200

100( 2h1 + 2h2 ) 2

Sag Vertical Curve (based on standard headlight criteria)

S > L

(

400 + 3.5 S A

)

AV 2 46.5

(

h +h 800 C− 1 2 A 2

L = 2S −

)

)

C = vertical clearance for overhead structure (overpass) located within 200 feet of the midpoint of the curve

Horizontal Curves Side friction factor (based on superelevation)

Spiral Transition Length

0.01e + f =

Ls =

V2 15 R

3.15V 3 RC

C = rate of increase of lateral acceleration [use 1 ft/sec3 unless otherwise stated] Sight Distance (to see around obstruction)

HSO = R

[

1 − cos

(

28.65 S R

)]

HSO = Horizontal sight line offset

CIVIL ENGINEERING

163

Horizontal Curve Formulas D = Degree of Curve, Arc Definition PC = Point of Curve (also called BC) PT = Point of Tangent (also called EC) PI = Point of Intersection I = Intersection Angle (also called Δ) Angle Between Two Tangents L = Length of Curve, from PC to PT T = Tangent Distance E = External Distance R = Radius LC = Length of Long Chord M = Length of Middle Ordinate c = Length of Sub-Chord d = Angle of Sub-Chord l = Curve Length for Sub-Chord R = 5729.58 D R=

LC 2 sin _ I 2i

T = R tan _ I 2i =

LC 2 cos _ I 2i

LATITUDES AND DEPARTURES

L = RI r = I 100 D 180 M = R 81 - cos _ I 2iB R cos _ I 2i E+R= R - M = cos _ I 2i R c = 2R sin _ d 2i l = Rd b r l 180 E = R=

1 - 1G cos _ I 2i

Deflection angle per 100 feet of arc length equals D 2

164

CIVIL ENGINEERING

+ Latitude

– Departure

+ Departure

– Latitude

Vertical Curve Formulas TANGENT OFFSET

PVI

x

BACK TANGENT

FORWARD TANGENT

L

E

y

PVT

g 2

PVC

g1 YPVC

DATUM

VERTICAL CURVE FORMULAS NOT TO SCALE

L

= Length of Curve (horizontal)

g2 = Grade of Forward Tangent

PVC = Point of Vertical Curvature

a = Parabola Constant

PVI = Point of Vertical Intersection

y = Tangent Offset

PVT = Point of Vertical Tangency

E = Tangent Offset at PVI

g1

= Grade of Back Tangent

r = Rate of Change of Grade

x

= Horizontal Distance from PVC to Point on Curve

xm = Horizontal Distance to Min/Max Elevation on Curve = -

g1 g1L 2a = g1 - g2

Tangent Elevation = YPVC + g1x and = YPVI + g2 (x – L/2) Curve Elevation = YPVC + g1x + ax2 = YPVC + g1x + [(g2 – g1)/(2L)]x2 y = ax 2

a=

g2 - g1 2L

E = ab Ll 2

2

r=

g2 - g1 L

EARTHWORK FORMULAS Average End Area Formula, V = L(A1 + A2)/2 Prismoidal Formula, V = L (A1 + 4Am + A2)/6, where Am L

= area of mid-section, and = distance between A1 and A2

Pyramid or Cone, V = h (Area of Base)/3

AREA FORMULAS Area by Coordinates: Area = [XA (YB – YN) + XB (YC – YA) + XC (YD – YB) + ... + XN (YA – YN – 1)] / 2 Trapezoidal Rule: Area = w c

h1 + hn + h2 + h3 + h4 + f + hn - 1m 2

Simpson’s 1/3 Rule: Area = = w > h1 + 2 e

! hk o + 4 e

n-2

k = 3, 5, f

! hk o k = 2, 4, f n-1

+ hn H 3

w = common interval n must be odd number of measurements w = common interval CIVIL ENGINEERING

165

Highway Pavement Design AASHTO Structural Number Equation SN = a1D1 + a2D2 +…+ anDn, where SN = structural number for the pavement ai = layer coefficient and Di = thickness of layer (inches).

Gross Axle Load kN

lb

Load Equivalency Factors Single

Tandem

Axles

Axles

kN

lb

Load Equivalency Factors Single

Tandem

Axles

Axles

4.45

1,000

0.00002

187.0

42,000

25.64

2.51

8.9

2,000

0.00018

195.7

44,000

31.00

3.00

17.8

4,000

0.00209

200.0

45,000

34.00

3.27

22.25

5,000

0.00500

204.5

46,000

37.24

3.55

26.7

6,000

0.01043

213.5

48,000

44.50

4.17

35.6

8,000

0.0343

222.4

50,000

52.88

4.86

44.5

10,000

0.0877

0.00688

231.3

52,000

5.63

53.4

12,000

0.189

0.0144

240.2

54,000

6.47

62.3

14,000

0.360

0.0270

244.6

55,000

6.93

66.7

15,000

0.478

0.0360

249.0

56,000

7.41

71.2

16,000

0.623

0.0472

258.0

58,000

8.45

80.0

18,000

1.000

0.0773

267.0

60,000

9.59

89.0

20,000

1.51

0.1206

275.8

62,000

10.84

97.8

22,000

2.18

0.180

284.5

64,000

12.22

106.8

24,000

3.03

0.260

289.0

65,000

12.96

111.2

25,000

3.53

0.308

293.5

66,000

13.73

115.6

26,000

4.09

0.364

302.5

68,000

15.38

124.5

28,000

5.39

0.495

311.5

70,000

17.19

133.5

30,000

6.97

0.658

320.0

72,000

19.16

142.3

32,000

8.88

0.857

329.0

74,000

21.32

151.2

34,000

11.18

1.095

333.5

75,000

22.47

155.7

35,000

12.50

1.23

338.0

76,000

23.66

160.0

36,000

13.93

1.38

347.0

78,000

26.22

169.0

38,000

17.20

1.70

356.0

80,000

28.99

178.0

40,000

21.08

2.08

k

166

Gross Axle Load

CIVIL ENGINEERING

d

lb

i hi

f lb h

Superpave PERFORMANCE-GRADED (PG) BINDER GRADING SYSTEM PERFORMANCE GRADE

PG 52 –10

–16

–22

AVERAGE 7-DAY MAXIMUM PAVEMENT DESIGN TEMPERATURE, °Ca MINIMUM PAVEMENT DESIGN TEMPERATURE, °Ca

–28

PG 64

PG 58 –34

–40

–46

–16

–22

–22 >–28 >–34 >–40 >–46 >–16 >–22 >–28 >–34 >–40 >–16 >–22 >–28 >–34 >–40

ORIGINAL BINDER FLASH POINT TEMP, T48: MINIMUM °C

230

VISCOSITY, ASTM D 4402: b MAXIMUM, 3 Pa-s (3,000 cP), TEST TEMP, °C

135

DYNAMIC SHEAR, TP5: c G*/sin δ , MINIMUM, 1.00 kPa TEST TEMPERATURE @ 10 rad/sec., °C

52

58

64

ROLLING THIN FILM OVEN (T240) OR THIN FILM OVEN (T179) RESIDUE MASS LOSS, MAXIMUM, %

1.00

DYNAMIC SHEAR, TP5: G*/sin δ , MAXIMUM, 2.20 kPa TEST TEMP @ 10 rad/sec. °C

52

58

64

PRESSURE AGING VESSEL RESIDUE (PP1) PAV AGING TEMPERATURE, °Cd DYNAMIC SHEAR, TP5: G*/sin δ , MINIMUM, 5,000 kPa TEST TEMP @ 10 rad/sec. °C

90

25

22

19

16

100

13

10

7

25

22

19

PHYSICAL HARDENING e CREEP STIFFNESS, TP1: f S, MAXIMUM, 300 MPa M-VALUE, MINIMUM, 0.300 TEST TEMP, @ 60 sec., °C DIRECT TENSION, TP3: f FAILURE STRAIN, MINIMUM, 1.0% TEST TEMP @ 1.0 mm/min, °C

100

16

13

28

25

22

19

16

REPORT

0

–6

–12

–18

–24

–30

–36

–6

–12

–18

–24

–30

–6

–12

–18

–24

–30

0

–6

–12

–18

–24

–30

–36

–6

–12

–18

–24

–30

–6

–12

–18

–24

–30

Federal Highway Administration Report FHWA-SA-95-03, "Background of Superpave Asphalt Mixture Design and Analysis," Nov. 1994.

CIVIL ENGINEERING

167

SUPERPAVE MIXTURE DESIGN: AGGREGATE AND GRADATION REQUIREMENTS PERCENT PASSING CRITERIA ( CONTROL POINTS ) STANDARD SIEVE, mm

50 37.5 25.0 19.0 12.5 9.5 2.36 0.075

MIXTURE DESIGNATIONS SUPERPAVE DESIG. (mm) NOMINAL MAX. SIZE (mm) MAXIMUM SIZE (mm)

NOMINAL MAXIMUM SIEVE SIZE ( mm )

9.5

12.5

100 90–100 32–67 2–10

19.0

100 90–100 28–58 2–10

25.0

100 90–100

23–49 2–8

TRAFFIC, MILLION EQUIV. SINGLE AXLE LOADS (ESALs)

37.5 100 90–100

100 90–100

19–45 1–7

19.0 19.0 25.0

25.0 25.0 37.5

37.5 37.5 50.0

ADDITIONAL REQUIREMENTS 0.6 – 1.2 4 0.80 15

15–41 0–6 FLAT AND ELONGATED PARTICLES

FINE AGGREGATE ANGULARITY

DEPTH FROM SURFACE > 100 m m ≤ 100 mm 55 / – 65 / – 75 / – 85 / 80 95 / 90 100 / 100 100 / 100

12.5 12.5 19.0

FINENESS-TO-EFFECTIVE ASPHALT RATIO, wt./wt. SHORT-TERM OVEN AGING AT 135°C , hr. TENSILE STRENGTH RATIO T283, min. TRAFFIC (FOR DESIGNATION) BASE ON, yr.

COARSE AGGREGATE ANGULARITY

< 0.3 W, then the bandwidth of xAM(t) is 2W. AM signals can be demodulated with an envelope detector or a synchronous demodulator.

Therefore, the impulse response h(t) and frequency response H(f) form a Fourier transform pair: h(t) ↔ H(f)

ELECTRICAL AND COMPUTER ENGINEERING

203

DSB (Double-Sideband Modulation)

FM (Frequency Modulation) The phase deviation is

xDSB(t) = Acm(t)cos(2πfct) If M(f) = 0 for |f | > W, then the bandwidth of m(t) is W and the bandwidth of xDSB(t) is 2W. DSB signals must be demodulated with a synchronous demodulator. A Costas loop is often used.

t

The frequency-deviation ratio is D=

SSB (Single-Sideband Modulation) Lower Sideband:

kF max m ^t h 2r W

where W is the message bandwidth. If D 1, (wideband FM) the 98% power bandwidth B is given by Carson’s rule: B ≅ 2(D + 1)W

In either case, if M(f) = 0 for |f | > W, then the bandwidth of xLSB(t) or of xUSB(t) is W. SSB signals can be demodulated with a synchronous demodulator or by carrier reinsertion and envelope detection.

The complete bandwidth of an angle-modulated signal is infinite.

Angle Modulation

Sampled Messages A lowpass message m(t) can be exactly reconstructed from uniformly spaced samples taken at a sampling frequency of fs = 1/Ts

xAng ^t h = Ac cos 7 2rfct + z ^t hA

The phase deviation φ(t) is a function of the message m(t). The instantaneous phase is zi ^t h = 2rfct + z ^t h radians

The instantaneous frequency is ~i ^t h = d zi ^t h = 2rfc + d z ^t h radians/s dt dt The frequency deviation is D~ ^t h = d z ^t h radians/s dt PM (Phase Modulation) The phase deviation is

z ^t h = kPm ^t h radians

204


A discriminator or a phase-lock loop can demodulate anglemodulated signals.

fs ≥ 2W where M(f ) = 0 for f > W The frequency 2W is called the Nyquist frequency. Sampled messages are typically transmitted by some form of pulse modulation. The minimum bandwidth B required for transmission of the modulated message is inversely proportional to the pulse length τ. B ∝ 1x Frequently, for approximate analysis B , 1 2x is used as the minimum bandwidth of a pulse of length τ.

Ideal-Impulse Sampling xd ^t h = m ^t h ! d _t - nTsi =

! m _ nTsi d _t - nTsi

n =+3

n =+3

n =-3

n =-3 k =+3

Xd _ f i = M _ f i ) fs ! d _ f - kfsi k =-3

= fs ! M _ f - kfsi k =+3 k =-3

The message m(t) can be recovered from xδ (t) with an ideal lowpass filter of bandwidth W. PAM (Pulse-Amplitude Modulation) Natural Sampling: A PAM signal can be generated by multiplying a message by a pulse train with pulses having duration τ and period Ts = 1/fs n =+3 t - nT xN ^t h = m ^t h ! P ; x s E = n =-3

t nT ! m ^t h P ; - s E x n =-3 n =+3

XN _ f i = xfs ! sinc _ kxfsi M _ f - kfsi k =+3 k =-3

The message m(t) can be recovered from xN(t) with an ideal lowpass filter of bandwidth W. PCM (Pulse-Code Modulation) PCM is formed by sampling a message m(t) and digitizing the sample values with an A/D converter. For an n-bit binary word length, transmission of a pulse-code-modulated lowpass message m(t), with M(f) = 0 for f > W, requires the transmission of at least 2nW binary pulses per second. A binary word of length n bits can represent q quantization levels: q = 2n The minimum bandwidth required to transmit the PCM message will be B ∝ nW = 2W log2 q


205

ANALOG FILTER CIRCUITS Analog filters are used to separate signals with different frequency content. The following circuits represent simple analog filters used in communications and signal processing.

First-Order Low-Pass Filters H ( jω )

0

H ( jω )

0

ω

ωc

H ( jω c ) =

1 2

RP =

V2 RP 1 = • V1 R1 1 + sRPC

R1R2 R1 + R2

H(s) =

RP =

206

1 RPC

V2 R2 1 = • V1 RS 1 + s L RS

RS = R1 + R2

H(s) =

ωc =

ωc =

RS L

I 2 RP 1 = • R sR I1 2 1+ PC

R1R2 R1 + R2


ωc =

1 RPC

ω

ωc

H ( jω c ) =

H (0)

Frequency Response

H(s) =

First-Order High-Pass Filters

1 2

H ( j ∞)

Frequency Response

H(s) =

sRS C V2 R2 = • V1 RS 1 + sRS C

RS = R1 + R2

H(s) =

RP =

RP =

1 RS C

s L RP V2 RP = • V1 R1 1 + s L RP

R1R2 R1 + R2

H(s) =

ωc =

ωc =

RP L

I2 RP s L RP = • I1 R2 1 + s L RP

R1R2 R1 + R2

ωc =

RP L

Band-Pass Filters

Band-Reject Filters

H ( jω )

H ( jω )

0 ωL ω 0 ω U ω

0 ωLω 0 ωU ω H( j ωL ) = H( j ωU ) =

1 2

H( j ω0 )

[

H ( jω L ) = H ( jωU ) = 1 −

1 2

]

H (0)

3-dB Bandwidth = BW = ωU − ω L

3-dB Bandwidth = BW = ωU − ω L

Frequency Response

Frequency Response

R1 v1 + _

C H(s) =

RP =

V2 1 s = • 2 V1 R1C s + s RPC + 1 LC

R1R2 R1 + R2

H ( jω 0 ) =

1

ω0 =

R2 R = P R1 + R2 R1

BW =

H(s) =

R2

+ v2 _

V2 R2 s 2 + 1 LC = • 2 V1 RS s + s RS C + 1 LC

LC

1

RS = R1 + R2

RP C

H (0) =

ω0 =

R2 R2 = R1 + R2 RS

BW =

1 LC 1 RS C

R1 R1

L

v1 +_

H(s) =

C R2

+ v2 _

v1 + _

V2 R2 s = • 2 L s + sRS L + 1 LC V1

RS = R1 + R2 R2 R H ( jω 0 ) = = 2 R1 + R2 RS

ω0 =

H(s) =

C L

+ R2 v2 _

V2 RP s 2 + 1 LC = • 2 V1 R1 s + sR P L + 1 LC

1 LC

R BW = S L

RP =

R1R2 R1 + R2

H (0) =

ω0 =

R2 R = P R1 + R2 R1

BW =

1 LC RP L


207

Phase-Lead Filter

Phase-Lag Filter

log|H(jω)|

log|H(jω)|

ω1

ωm

ω

ω2

ω1

∠ H( jω )

ωm

ω2

ω

ωm

ω2

ω

∠ H( jω )

0

φm

φm

0

ω1

ωm

ω2

ω

ω1

Frequency Response

Frequency Response

C R1

_ v1 +

=

R1R2 R1 + R2

ω1 =

φm = arctan = arctan

=

1 R1C

ω2 =

1 RPC

max{∠ H ( jω m )} = φ m ω2 ω1 − arctan ω1 ω2 ω 2 − ω1 2ω m

H (0) =

|H ( j ω m ) | =

ω1 ω2

H ( j∞) = 1


v2 −

1 + s ω2 1 + s ω1

RS = R1 + R2

ω1 =

φm = arctan

1 RS C

ω2 =

1 R2 C

min{∠ H ( jω m )} = φ m

ω m = ω1ω 2

= arctan

RP ω1 = R1 ω2

C

+

V2 1 + sR2C = V1 1 + sRS C

H(s) =

ω1 1 + s ω1 • ω2 1+ s ω2

ω m = ω1ω 2

208

+ v2 _

R2

R2

v1 + −

V2 RP 1 + sR1C = • V1 R1 1 + sRPC

H(s) =

RP =

R1

ω1 ω2 − arctan ω2 ω1 ω1 − ω 2 2ω m

H(0) = 1 |H ( jω m ) | = H( j∞) =

ω1 ω2

R2 ω = 1 RS ω2

OPERATIONAL AMPLIFIERS

Built-in potential (contact potential) of a p-n junction:

Ideal v2 v0 = A(v1 – v2) v0 v1 where A is large (> 104), and v1 – v2 is small enough so as not to saturate the amplifier. For the ideal operational amplifier, assume that the input currents are zero and that the gain A is infinite so when operating linearly v2 – v1 = 0. For the two-source configuration with an ideal operational amplifier, R2

NaNd V0 = kT q ln n 2 i

Thermal voltage VT = kT q . 0.026V at 300K Na = acceptor concentration, Nd = donor concentration, T = temperature (K), and k = Boltzmann’s Constant = 1.38 × 10–23 J/K Capacitance of abrupt p – n junction diode C ^V h = C0

R1

va

v 0 =-

vb

v0

R2 R2 v d1 nv R1 a + + R1 b

If va = 0, we have a non-inverting amplifier with R v 0 = d1 + 2 n v b R1

1 - V/Vbi , where:

C0 = junction capacitance at V = 0, V = potential of anode with respect to cathode, and Vbi = junction contact potential. Resistance of a diffused layer is R = Rs (L/W), where: Rs = sheet resistance = ρ/d in ohms per square ρ = resistivity, d = thickness, L = length of diffusion, and W = width of diffusion.

If vb = 0, we have an inverting amplifier with v 0 =-

R2 v R1 a

SOLID-STATE ELECTRONICS AND DEVICES Conductivity of a semiconductor material: σ = q (nμn + pμp), where μn μp n p q

TABULATED CHARACTERISTICS FOR: Diodes Bipolar Junction Transistors (BJT) N-Channel JFET and MOSFETs Enhancement MOSFETs are on the following pages.

≡ electron mobility, ≡ hole mobility, ≡ electron concentration, ≡ hole concentration, and ≡ charge on an electron (1.6 × 10−19C).

Doped material: p-type material; pp ≈ Na n-type material; nn ≈ Nd Carrier concentrations at equilibrium (p)(n) = ni2, where ni ≡ intrinsic concentration.


209

DIODES

iD

vD

vB

iD

−v z

iD vD

]

[

iD ≈ I s e (v D ηVT ) – 1 where Is = saturation current η = emission coefficient, typically 1 for Si VT = thermal voltage = kT q

vD

vB = breakdown voltage

(Zener Diode)

iD

Shockley Equation

(0.5 to 0.7)V

− C

+ vD

Mathematical I – V Relationship

iD

iD

(Junction Diode)

A

Piecewise-Linear Approximation of the I – V Relationship

Ideal I – V Relationship

Device and Schematic Symbol

Same as above.

−vZ

vD (0.5 to 0.7)V

A

C

+ vD −

vZ

= Zener voltage

NPN Bipolar Junction Transistor (BJT) Mathematical Large-Signal (DC) Relationships Equivalent Circuit

Schematic Symbol

Low Frequency:

Active Region: C

iC iB

iE iC iC α

B

iE

iC IS VT

E

= iB + iC = β iB = α iE = β /( β + 1) ≈ IS e (V BE VT ) = emitter saturation current = thermal voltage

base emitter junction forward biased; base collector juction reverse biased

C

iB

gm ≈ ICQ/VT rπ ≈ β/gm,

iC iB

B V BE

Low-Frequency Small-Signal (AC) Equivalent Circuit

iE E

ro = ∂ vCE ∂ ic

[ ]

Note: These relationships are valid in the active mode of operation.

ib (t)

both junctions forward biased

C

iC

VCE sat

Qpoint

B

r

iB

i c(t)

ie (t) VBE sat E

iC

iB

iE

Cutoff Region : Same as for NPN with current directions and voltage polarities reversed.

B

both junctions reverse biased

C

B

iE E

PNP – Transistor

210


ro

g mv be

B

C

VA I CQ

where ICQ = dc collector current at the Q point VA = Early voltage

Saturation Region : NPN – Transistor

≈

E Same as NPN with current directions and voltage polarities reversed

E Same as for NPN.

C

Schematic Symbol N-CHANNEL JFET D

N-Channel Junction Field Effect Transistors (JFETs) and Depletion MOSFETs (Low and Medium Frequency) Mathematical Relationships Small-Signal (AC) Equivalent Circuit

Cutoff Region: vGS < Vp iD = 0

gm =

2 I DSS I D

Triode Region: vGS > Vp and vGD > Vp iD = (IDSS /Vp2)[2vDS (vGS – Vp) – vDS2 ]

G iS

P-CHANNEL JFET D iD G

where IDSS = drain current with vGS = 0 (in the saturation region) = KV p2, K = conductivity factor, and Vp = pinch-off voltage.

iS

iD (t)

G

D

+

Saturation Region: vGS > Vp and vGD < V p iD = IDSS (1 – vGS /Vp)2

S

in saturation region

Vp

iD

v gs

g mv gs

rd

v ds

S

where rd =

∂vds ∂id

Q point

S

N-CHANNEL DEPLETION MOSFET (NMOS) D iD G B iS S

SIMPLIFIED SYMBOL D iD G iS S

P-CHANNEL DEPLETION MOSFET (PMOS)

Same as for N-Channel with current directions and voltage polarities reversed.

Same as for N-Channel.

D iD

G

B iS S

SIMPLIFIED SYMBOL D iD

G iS S


211

Schematic Symbol N-CHANNEL ENHANCEMENT MOSFET (NMOS) D

Enhancement MOSFET (Low and Medium Frequency) Mathematical Relationships Small-Signal (AC) Equivalent Circuit Cutoff Region : vGS < Vt iD = 0

gm = 2K(vGS – Vt ) in saturation region G

Triode Region : vGS > Vt and vGD > Vt 2 iD = K [2vDS (v GS – Vt) – vDS ]

iD G

vgs

D

g m vgs

B

Saturation Region : vGS > Vt and vGD < Vt 2

is S SIMPLIFIED SYMBOL D

iD = K (vGS – Vt) where K = conductivity factor V t = threshold voltage

iD

S

where

rd =

∂vds ∂id

Qpoint

G is S P-CHANNEL ENHANCEMENT MOSFET (PMOS)

Same as for N-channel with current directions and voltage polarities reversed.

D

iD G B is S SIMPLIFIED SYMBOL D iD G is S

212


Same as for N-channel.

rd

+ vds −

NUMBER SYSTEMS AND CODES An unsigned number of base-r has a decimal equivalent D defined by n

m

k=0

i=1

D = ! akr k + ! air- i, where ak = the (k + 1) digit to the left of the radix point and ai = the ith digit to the right of the radix point.

LOGIC OPERATIONS AND BOOLEAN ALGEBRA Three basic logic operations are the “AND (•),” “OR (+),” and “Exclusive-OR ⊕” functions. The definition of each function, its logic symbol, and its Boolean expression are given in the following table. Function

Binary Number System In digital computers, the base-2, or binary, number system is normally used. Thus the decimal equivalent, D, of a binary number is given by D = ak 2k + ak – 12k – 1 + … + a0 + a–1 2–1 + … Since this number system is so widely used in the design of digital systems, we use a short-hand notation for some powers of two: 210 = 1,024 is abbreviated “k” or “kilo” 220 = 1,048,576 is abbreviated “M” or “mega” Signed numbers of base-r are often represented by the radix complement operation. If M is an N-digit value of base-r, the radix complement R(M) is defined by

Inputs AB

C = A• B

C= A+ B

C= A⊕B

00

0

0

0

01

0

1

1

10

0

1

1

11

1

1

0

As commonly used, A AND B is often written AB or A•B. The not operator inverts the sense of a binary value (0 → 1, 1 → 0)

R(M) = r N – M The 2’s complement of an N-bit binary integer can be written

C=Ā

A

2’s Complement (M) = 2N – M This operation is equivalent to taking the 1’s complement (inverting each bit of M) and adding one. The following table contains equivalent codes for a four-bit binary value. HexaBinary Decimal Octal decimal Base-2 Base-10 Base-8 Base-16 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0 1 2 3 4 5 6 7 8 9 A B C D E F

0 1 2 3 4 5 6 7 10 11 12 13 14 15 16 17

LOGIC SYMBOL

DeMorgan’s Theorems first theorem: A + B = A : B

BCD Code 0 1 2 3 4 5 6 7 8 9 -------------

Gray Code 0000 0001 0011 0010 0110 0111 0101 0100 1100 1101 1111 1110 1010 1011 1001 1000

second theorem: A : B = A + B These theorems define the NAND gate and the NOR gate. Logic symbols for these gates are shown below. NAND Gates: A • B = A + B A

A NAND

C

NAND

C

B

B

NOR Gates: A + B = A • B A

A NOR B

C

NOR

C

B


213

FLIP-FLOPS A flip-flop is a device whose output can be placed in one of two states, 0 or 1. The flip-flop output is synchronized with a clock (CLK) signal. Qn represents the value of the flipflop output before CLK is applied, and Qn+1 represents the output after CLK has been applied. Three basic flip-flops are described below. RS Flip-Flop S

JK Flip-Flop J

Q

CLK R

SR

D Flip-Flop

Q

CLK Q

CLK Q

K

Qn+1

Q

D

JK

Q

Qn+1

D Qn+1

00 Qn no change 01 0

00 Qn no change 01 0

10 1

10 1

11 x invalid

11 Qn toggle

0 1

Qn

Qn+1

S

R

J

K

D

0 0

0 1 0

x 0 1

0 1 x

x x

1

0 1 0

1

0 1 0

1

1

x

0

x

0

1

Switching Function Terminology Minterm, mi – A product term which contains an occurrence of every variable in the function. Maxterm, Mi – A sum term which contains an occurrence of every variable in the function. Implicant – A Boolean algebra term, either in sum or product form, which contains one or more minterms or maxterms of a function. Prime Implicant – An implicant which is not entirely contained in any other implicant. Essential Prime Implicant – A prime implicant which contains a minterm or maxterm which is not contained in any other prime implicant.


F(ABCD) = Σm(h, i, j,…) = mh + mi + mj + … A function can be described as a product of maxterms using the notation G(ABCD) = ΠM(h, i, j,…) = M h • Mi • Mj • … A function represented as a sum of minterms only is said to be in canonical sum of products (SOP) form. A function represented as a product of maxterms only is said to be in canonical product of sums (POS) form. A function in canonical SOP form is often represented as a minterm list, while a function in canonical POS form is often represented as a maxterm list. A Karnaugh Map (K-Map) is a graphical technique used to represent a truth table. Each square in the K-Map represents one minterm, and the squares of the K-Map are arranged so that the adjacent squares differ by a change in exactly one variable. A four-variable K-Map with its corresponding minterms is shown below. K-Maps are used to simplify switching functions by visually identifying all essential prime implicants.

0 1

Composite Flip-Flop State Transition

214

A function can be described as a sum of minterms using the notation

Four-variable Karnaugh Map CD AB 00

00

01

11

10

m0

m1

m3

m2

01

m4

m5

m7

m6

11

m12

m13

m15

m14

10

m8

m9

m11

m10

INDUSTRIAL ENGINEERING LINEAR PROGRAMMING The general linear programming (LP) problem is: Maximize Z = c1x1 + c2x2 + … + cnxn Subject to: a11x1 + a12x2 + … + a1nxn ≤ b1 a21x1 + a22x2 + … + a2nxn ≤ b2 h am1x1 + am2x2 + … + amnxn ≤ bm x1, … , xn ≥ 0 An LP problem is frequently reformulated by inserting nonnegative slack and surplus variables. Although these variables usually have zero costs (depending on the application), they can have non-zero cost coefficients in the objective function. A slack variable is used with a "less than" inequality and transforms it into an equality. For example, the inequality 5x1 + 3x2 + 2x3 ≤ 5 could be changed to 5x1 + 3x2 + 2x3 + s1 = 5 if s1 were chosen as a slack variable. The inequality 3x1 + x2 – 4x3 ≥ 10 might be transformed into 3x1 + x2 – 4x3 – s2 = 10 by the addition of the surplus variable s2. Computer printouts of the results of processing an LP usually include values for all slack and surplus variables, the dual prices, and the reduced costs for each variable. Dual Linear Program Associated with the above linear programming problem is another problem called the dual linear programming problem. If we take the previous problem and call it the primal problem, then in matrix form the primal and dual problems are respectively: Primal Maximize Z = cx Subject to: Ax ≤ b x≥0

Dual Minimize W = yb Subject to: yA ≥ c y≥0

It is assumed that if A is a matrix of size [m × n], then y is a [1 × m] vector, c is a [1 × n] vector, b is an [m × 1] vector, and x is an [n × 1] vector. Network Optimization Assume we have a graph G(N, A) with a finite set of nodes N and a finite set of arcs A. Furthermore, let N = {1, 2, … , n} xij = flow from node i to node j cij = cost per unit flow from i to j uij = capacity of arc (i, j) bi = net flow generated at node i We wish to minimize the total cost of sending the available supply through the network to satisfy the given demand. The minimal cost flow model is formulated as follows:

n

n

Minimize Z = ! ! cij xij i=1j=1

subject to n

n

j=1

j=1

! xij - ! x ji = bi for each node i ! N

and

0 ≤ xij ≤ uij for each arc (i, j)∈A The constraints on the nodes represent a conservation of flow relationship. The first summation represents total flow out of node i, and the second summation represents total flow into node i. The net difference generated at node i is equal to bi. Many models, such as shortest-path, maximal-flow, assignment and transportation models can be reformulated as minimal-cost network flow models.

STATISTICAL QUALITY CONTROL Average and Range Charts n

A2

2 3 4 5 6 7 8 9 10

1.880 1.023 0.729 0.577 0.483 0.419 0.373 0.337 0.308

Xi n k R

D3 0 0 0 0 0 0.076 0.136 0.184 0.223

D4 3.268 2.574 2.282 2.114 2.004 1.924 1.864 1.816 1.777

= an individual observation = the sample size of a group = the number of groups = (range) the difference between the largest and smallest observations in a sample of size n. X=

X1 + X2 + f + Xn n

X=

X1 + X 2 + f + X n k

R=

R1 + R2 + f + Rk k

The R Chart formulas are: CLR = R UCLR = D4R LCLR = D3R

INDUSTRIAL ENGINEERING

215

The X Chart formulas are: CLX = X UCLX = X + A2R LCLX = X - A2R Standard Deviation Charts n 2 3 4 5 6 7 8 9 10

A3 2.659 1.954 1.628 1.427 1.287 1.182 1.099 1.032 0.975

B3 0 0 0 0 0.030 0.119 0.185 0.239 0.284

B4 3.267 2.568 2.266 2.089 1.970 1.882 1.815 1.761 1.716

CLX = X

Actual Capability n LSL USL - n m PCRk = Cpk = min c , 3v 3v Potential Capability (i.e., Centered Process) PCR = Cp = USL - LSL , where 6v

LCLX = X - A3S

QUEUEING MODELS

UCLS = B4S

Definitions Pn = probability of n units in system, L = expected number of units in the system, Lq = expected number of units in the queue, W = expected waiting time in system, Wq = expected waiting time in queue, λ = mean arrival rate (constant), mu = effective arrival rate, μ = mean service rate (constant), ρ = server utilization factor, and s = number of servers.

CLS = S LCLS = B3S Approximations The following table and equations may be used to generate initial approximations of the items indicated. c4 0.7979 0.8862 0.9213 0.9400 0.9515 0.9594 0.9650 0.9693 0.9727

d2 1.128 1.693 2.059 2.326 2.534 2.704 2.847 2.970 3.078

d3 0.853 0.888 0.880 0.864 0.848 0.833 0.820 0.808 0.797

vt = R d2 vt = S c4 vR = d3 vt vS = vt 1 - c42, where vt = an estimate of σ, σR = an estimate of the standard deviation of the ranges of the samples, and σS = an estimate of the standard deviation of the standard deviations of the samples.

216

PROCESS CAPABILITY

μ and σ are the process mean and standard deviation, respectively, and LSL and USL are the lower and upper specification limits, respectively.

UCLX = X + A3S

n 2 3 4 5 6 7 8 9 10

Tests for Out of Control 1. A single point falls outside the (three sigma) control limits. 2. Two out of three successive points fall on the same side of and more than two sigma units from the center line. 3. Four out of five successive points fall on the same side of and more than one sigma unit from the center line. 4. Eight successive points fall on the same side of the center line.


Kendall notation for describing a queueing system: A/B/s/M A = the arrival process, B = the service time distribution, s = the number of servers, and M = the total number of customers including those in service. Fundamental Relationships L = λW Lq = λWq W = Wq + 1/μ ρ = λ/(sμ)

Single Server Models (s = 1) Poisson Input—Exponential Service Time: M = ∞ P0 = 1 – λ/μ = 1 – ρ Pn = (1 – ρ)ρn = P0ρn L = ρ/(1 – ρ) = λ/(μ – λ) Lq = λ2/[μ (μ– λ)] W = 1/[μ (1 – ρ)] = 1/(μ – λ) Wq = W – 1/μ = λ/[μ (μ – λ)] Finite queue: M < ∞ mu = m _1 - Pni P0 = (1 – ρ)/(1 – ρM+1) Pn = [(1 – ρ)/(1 – ρM+1)]ρn L = ρ/(1 – ρ) – (M + 1)ρM+1/(1 – ρM+1) Lq = L – (1 – P0) Poisson Input—Arbitrary Service Time Variance σ2 is known. For constant service time, σ2 = 0. P0 = 1 – ρ Lq = (λ2σ2 + ρ2)/[2 (1 – ρ)] L = ρ + Lq Wq = L q / λ W = Wq + 1/μ

Calculations for P0 and Lq can be time consuming; however, the following table gives formulas for 1, 2, and 3 servers. s 1 2 3

P0 1–ρ (1 – ρ)/(1 + ρ) 2(1 − ρ ) 2 + 4ρ + 3ρ 2

mm t P0 c n

1. Random Variate Generation The linear congruential method of generating pseudorandom numbers Ui between 0 and 1 is obtained using Zn = (aZn–1+C) (mod m) where a, C, m, and Z0 are given nonnegative integers and where Ui = Zi /m. Two integers are equal (mod m) if their remainders are the same when divided by m. 2. Inverse Transform Method If X is a continuous random variable with cumulative distribution function F(x), and Ui is a random number between 0 and 1, then the value of Xi corresponding to Ui can be calculated by solving Ui = F(xi) for xi. The solution obtained is xi = F–1(Ui), where F–1 is the inverse function of F(x). F(x) 1 U1

U2 X X2

s

=

2

s+1

P0s t 2 s! ^1 - th

Pn = P0 ^m nh /n! n

0 # n # s

Pn = P0 ^m nh / _ s!s n - si n $ s Wq = Lq m n

W = Wq + 1 n L = Lq + m n

X2

FORECASTING Moving Average n

dtt =

s! ^1 - th

0

Inverse Transform Method for Continuous Random Variables

s

Lq =

9ρ 4 2 + 2ρ − ρ 2 − 3ρ 3

SIMULATION

Poisson Input—Erlang Service Times, σ2 = 1/(kμ2) Lq = [(1 + k)/(2k)][(λ2)/(μ (μ– λ))] = [λ2/(kμ2) + ρ2]/[2(1 – ρ)] Wq = [(1 + k)/(2k)]{λ /[μ (μ – λ)]} W = Wq + 1/μ Multiple Server Model (s > 1) Poisson Input—Exponential Service Times R V- 1 n s S W m m cn m Ss - 1 cn m W 1 P0 = S ! + s! W n ! f p m 1 - sn W Sn = 0 T X n s s - 1 ^ sth ^ sth H = 1 > ! n! + s! ^1 - th n=0

Lq ρ /(1 – ρ) 2ρ 3/(1 – ρ 2) 2

! dt - i

i=1

n

, where

dtt = forecasted demand for period t, dt – i = actual demand for ith period preceding t, and n = number of time periods to include in the moving average. Exponentially Weighted Moving Average dtt = adt 1 + ^1 - ah dtt 1, where -

dtt α

-

= forecasted demand for t, and = smoothing constant, 0 ≤ α ≤ 1


217

LINEAR REGRESSION

C = T2 N

Least Squares t , where y = at + bx

i=1j=1

SStreatments = ! `Ti2 ni j - C k

t , y - intercept: at = y - bx and slope: bt = Sxy Sxx,

i=1

SSerror = SStotal - SStreatments

Sxy = ! xi yi - _1 ni d ! xi n d ! yi n, n

n

i=1

i=1

n

See One-Way ANOVA table later in this chapter.

i=1

Sxx = ! xi2 - _1 ni d ! xi n , 2

n

n

i=1

i=1

RANDOMIZED BLOCK DESIGN The experimental material is divided into n randomized blocks. One observation is taken at random for every treatment within the same block. The total number of observations is N = nk. The total value of these observations is equal to T. The total value of observations for treatment i is Ti. The total value of observations in block j is Bj.

n = sample size, y = _1 ni d ! yi n, and n

i=1

x = _1 ni d ! xi n . n

i=1

C = T2 N

Standard Error of Estimate

k

SxxSyy = MSE, where Sxx ^n - 2h

Syy = ! yi2 - _1 ni d ! yi n n

n

i=1

i=1

i=1j=1

SSblocks = ! ` B 2j k j - C n

j=1

2

SStreatments = ! `Ti2 nj - C k

i=1

SSerror = SStotal - SSblocks - SStreatments

Confidence Interval for a e 1 + x o MSE n Sxx 2

at ! ta

2, n - 2

See Two-Way ANOVA table later in this chapter.

Confidence Interval for b bt ! ta

2, n - 2

MSE Sxx

Sample Correlation Coefficient Sxy SxxSyy

r=

ONE-WAY ANALYSIS OF VARIANCE (ANOVA) Given independent random samples of size ni from k populations, then: ! ! _ xij - x i k

ni

2

= ! ! _ xij - xi i + ! ni _ xi - x i or ni

2n FACTORIAL EXPERIMENTS Factors: X1, X2, …, Xn Levels of each factor: 1, 2 (sometimes these levels are represented by the symbols – and +, respectively) r = number of observations for each experimental condition (treatment), Ei = estimate of the effect of factor Xi, i = 1, 2, …, n, Eij = estimate of the effect of the interaction between factors Xi and Xj, Y ik = average response value for all r2n – 1 observations having Xi set at level k, k = 1, 2, and km

i=1j=1

k

2

i=1j=1

k

2

i=1

Y ij = average response value for all r2n – 2 observations having Xi set at level k, k = 1, 2, and Xj set at level m, m = 1, 2. Ei = Y i2 - Y i1 Eij =

Let T be the grand total of all N = Σini observations and Ti be the total of the ni observations of the ith sample.


cY ij - Y ij m - cY ij - Y ij m 22

SStotal = SSerror + SStreatments

218

n

SStotal = ! ! xij2 - C

2 Sxy

Se2 =

ni

k

SStotal = ! ! xij2 - C

21

12

2

11

ANALYSIS OF VARIANCE FOR 2n FACTORIAL DESIGNS Main Effects Let E be the estimate of the effect of a given factor, let L be the orthogonal contrast belonging to this effect. It can be proved that E=

L 2n - 1

where SSi is the sum of squares due to factor Xi, SSij is the sum of squares due to the interaction of factors Xi and Xj, and so on. The total sum of squares is equal to 2 m r SStotal = ! ! Yck2 - T N c = 1k = 1

where Yck is the kth observation taken for the cth experimental condition, m = 2n, T is the grand total of all observations, and N = r2n.

m

L = ! a^chY ^ch c=1

2 SSL = rLn , where 2

m = number of experimental conditions (m = 2n for n factors), a(c) = –1 if the factor is set at its low level (level 1) in experimental condition c, a(c) = +1 if the factor is set at its high level (level 2) in experimental condition c, r = number of replications for each experimental condition Y ^ch = average response value for experimental condition c, and SSL = sum of squares associated with the factor. Interaction Effects Consider any group of two or more factors. a(c) = +1 if there is an even number (or zero) of factors in the group set at the low level (level 1) in experimental condition c = 1, 2, …, m a(c) = –1 if there is an odd number of factors in the group set at the low level (level 1) in experimental condition c = 1, 2, …, m It can be proved that the interaction effect E for the factors in the group and the corresponding sum of squares SSL can be determined as follows: E=

L 2n - 1

RELIABILITY If Pi is the probability that component i is functioning, a reliability function R(P1, P2, …, Pn) represents the probability that a system consisting of n components will work. For n independent components connected in series, R _ P1, P2, fPni = % Pi n

i=1

For n independent components connected in parallel, R _ P1, P2, fPni = 1 - % _1 - Pi i n

i=1

LEARNING CURVES The time to do the repetition N of a task is given by TN = KN s, where K = constant, and s = ln (learning rate, as a decimal)/ln 2; or, learning rate = 2s. If N units are to be produced, the average time per unit is given by Tavg =

K 8^ N + 0.5h^1 + sh - 0.5^1 + shB N ^1 + sh

INVENTORY MODELS For instantaneous replenishment (with constant demand rate, known holding and ordering costs, and an infinite stockout cost), the economic order quantity is given by

m

L = ! a^chY ^ch c=1

2 SSL = rLn 2

Sum of Squares of Random Error The sum of the squares due to the random error can be computed as SSerror = SStotal – ΣiSSi – ΣiΣjSSij – … – SS12 …n

EOQ =

2AD , where h

A = cost to place one order, D = number of units used per year, and h = holding cost per unit per year. Under the same conditions as above with a finite replenishment rate, the economic manufacturing quantity is given by EMQ =

2AD , where h _1 - D Ri

R = the replenishment rate.


219

ERGONOMICS NIOSH Formula Recommended Weight Limit (pounds) = 51(10/H)(1 – 0.0075|V – 30|)(0.82 + 1.8/D)(1 – 0.0032A)(FM)(CM) where H = horizontal distance of the hand from the midpoint of the line joining the inner ankle bones to a point projected on the floor directly below the load center, in inches V = vertical distance of the hands from the floor, in inches D = vertical travel distance of the hands between the origin and destination of the lift, in inches A = asymmetry angle, in degrees FM = frequency multiplier (see table) CM = coupling multiplier (see table)

Frequency Multiplier Table F, min 0.2 0.5 1 2 3 4

–1

5 6 7 8 9 10 11 12 13 14 15

220


≤ 8 hr /day V< 30 in. V ≥ 30 in. 0.85 0.81 0.75 0.65 0.55 0.45

≤ 2 hr /day V < 30 in. V ≥ 30 in. 0.95 0.92 0.88 0.84 0.79 0.72

0.35 0.27 0.22 0.18

0.60 0.50 0.42 0.35 0.30 0.26

0.15 0.13

0.23 0.21 0.00

≤ 1 hr /day V < 30 in. V ≥ 30 in. 1.00 0.97 0.94 0.91 0.88 0.84 0.80 0.75 0.70 0.60 0.52 0.45 0.41 0.37 0.34 0.31 0.28

Coupling Multiplier (CM) Table (Function of Coupling of Hands to Load) Container Optimal Design Opt. Handles Not or Cut-outs GOOD

Not POOR

Loose Part / Irreg. Object Comfort Grip Not GOOD

Flex Fingers 90 Degrees FAIR

Coupling GOOD FAIR POOR

V < 30 in. or 75 cm 1.00 0.95 0.90

Not POOR

V ≥ 30 in. or 75 cm

Biomechanics of the Human Body Basic Equations Hx + Fx = 0 Hy + Fy = 0 Hz + W+ Fz = 0 THxz + TWxz + TFxz = 0 THyz + TWyz + TFyz = 0 THxy + TFxy = 0

W

The coefficient of friction μ and the angle α at which the floor is inclined determine the equations at the foot. Fx = μFz With the slope angle α Fx = αFzcos α Of course, when motion must be considered, dynamic conditions come into play according to Newton's Second Law. Force transmitted with the hands is counteracted at the foot. Further, the body must also react with internal forces at all points between the hand and the foot.


221

PERMISSIBLE NOISE EXPOSURE (OSHA)

Standard Time Determination

Noise dose (D) should not exceed 100%. C D = 100% # ! i Ti

ST = NT × AF where NT = normal time, and AF = allowance factor.

where Ci

=

Ti

=

! Ci =

time spent at specified sound pressure level, SPL, (hours) time permitted at SPL (hours) 8 (hours)

For 80 ≤ SPL ≤ 130 dBA, Ti = 2c

105 - SPL m 5

(hours)

If D > 100%, noise abatement required. If 50% ≤ D ≤ 100%, hearing conservation program required. Note: D = 100% is equivalent to 90 dBA time-weighted average (TWA). D = 50% equivalent to TWA of 85 dBA. Hearing conservation program requires: (1) testing employee hearing, (2) providing hearing protection at employee’s request, and (3) monitoring noise exposure. Exposure to impulsive or impact noise should not exceed 140 dB sound pressure level (SPL).

Case 1: Allowances are based on the job time. AFjob = 1 + Ajob Ajob = allowance fraction (percentage/100) based on job time. Case 2: Allowances are based on workday. AFtime = 1/(1 – Aday) Aday = allowance fraction (percentage/100) based on workday. Plant Location The following is one formulation of a discrete plant location problem. Minimize m

n

n

z = ! ! cij yij + ! fjx j i=1j=1

j=1

subject to

FACILITY PLANNING Equipment Requirements n PijTij Mj = ! where i = 1 Cij Mj = number of machines of type j required per production period, Pij = desired production rate for product i on machine j, measured in pieces per production period, Tij = production time for product i on machine j, measured in hours per piece, Cij = number of hours in the production period available for the production of product i on machine j, and n = number of products. People Requirements n PijTij A j = ! ij , where i=1 C

Aj = number of crews required for assembly operation j, Pij = desired production rate for product i and assembly operation j (pieces per day), Tij = standard time to perform operation j on product i (minutes per piece), Cij = number of minutes available per day for assembly operation j on product i, and n = number of products.

222


m

! yij # mx j,

j = 1 , f, n

! yij = 1,

i = 1 , f, m

i=1 n

j=1

yij $ 0, for all i, j

x j = ^0, 1h, for all j, where m = number of customers, n = number of possible plant sites, yij = fraction or proportion of the demand of customer i which is satisfied by a plant located at site j; i = 1, …, m; j = 1, …, n, xj = 1, if a plant is located at site j, xj = 0, otherwise, cij = cost of supplying the entire demand of customer i from a plant located at site j, and fj = fixed cost resulting from locating a plant at site j. Material Handling Distances between two points (x1, y1) and (x2, y2) under different metrics: Euclidean:

D = ^ x1 - x2h + _ y1 - y2i 2

Rectilinear (or Manhattan): D = x1 - x2 + y1 - y2

2

Chebyshev (simultaneous x and y movement): D = max _ x1 - x2 , y1 - y2 i

TAYLOR TOOL LIFE FORMULA

Line Balancing

V = speed in surface feet per minute, T = tool life in minutes, and C, n = constants that depend on the material and on the tool.

Nmin = bOR # ! ti OT l i = Theoretical minimum number of stations Idle Time/Station = CT - ST Idle Time/Cycle = ! ^CT - ST h Percent Idle Time =

Idle Time/Cycle # 100, where Nactual # CT

CT = cycle time (time between units), OT = operating time/period, OR = output rate/period, ST = station time (time to complete task at each station), ti = individual task times, and N = number of stations.

VT n = C, where

WORK SAMPLING FORMULAS p _1 - pi and R = Za D = Za 2 n

2

1-p pn , where

p = proportion of observed time in an activity, D = absolute error, R = relative error (R = D/p), and n = sample size.

Job Sequencing Two Work Centers—Johnson’s Rule 1. Select the job with the shortest time, from the list of jobs, and its time at each work center. 2. If the shortest job time is the time at the first work center, schedule it first, otherwise schedule it last. Break ties arbitrarily. 3. Eliminate that job from consideration. 4. Repeat 1, 2, and 3 until all jobs have been scheduled.

CRITICAL PATH METHOD (CPM) dij = duration of activity (i, j), CP = critical path (longest path), T = duration of project, and T = ! dij _i, j i ! CP

PERT (aij, bij, cij) = (optimistic, most likely, pessimistic) durations for activity (i, j), μij = mean duration of activity (i, j), σij = standard deviation of the duration of activity (i, j), μ = project mean duration, and σ = standard deviation of project duration. aij + 4bij + cij nij = 6 cij - aij vij = 6 n = ! nij _i, j i ! CP

2

v =

! vij2

_i, j i ! CP


223

Source of Variation

Between Treatments

k–1

SStreatments

Error

N–k

SSerror

Total

N–1

SStotal

Source of Variation

224

ONE-WAY ANOVA TABLE Degrees of Sum of Mean Square Freedom Squares SStreatments k −1

MSE =

k–1

SStreatments

Between Blocks

n–1

SSblocks

Error

(k − 1 )(n − 1 )

SSerror

Total

N–1

SStotal

MST =

F

SStreatments k −1

MST MSE

SS blocks n −1

MSB MSE

MSB =

MSE =

MST MSE

SS error N −k

TWO-WAY ANOVA TABLE Degrees of Sum of Mean Square Freedom Squares

Between Treatments


MST =

F

SS error (k − 1 )(n − 1 )

PROBABILITY AND DENSITY FUNCTIONS: MEANS AND VARIANCES Variable Equation Mean Variance Binomial Coefficient Binomial

Hyper Geometric

( nx ) = x!(nn−! x)! n b(x; n , p ) = ( ) p (1 − p ) x

λ

λ

1/p

(1 – p)/p2

)

r/p

r (1 – p)/p2

n! p1x1 … pkxk x1! , …, xk !

npi

npi (1 – pi)

(a + b)/2

(b – a)2/12

x α − 1e − x β ; α > 0, β > 0 β α Γ (α )

αβ

αβ 2

1 −x β e β

β

β2

f ( y ; r, p) =

f ( x1, …, xk) =

N −r n−x

(Nn )

λ x e −λ x!

(

y + r −1 r p (1 − p ) y r −1

f(x) = 1/(b – a) f (x ) =

f (x ) = f (x ) =

Weibull

Triangular

r (N − r )n(N − n ) N 2 (N − 1)

( )

g(x; p) = p (1 – p)x–1

Exponential

Normal

nr N

f (x; λ ) =

Uniform Gamma

np(1 – p)

()

Geometric

Multinomial

np

r h(x; n, r, N ) = x

Poisson

Negative Binomial

n− x

x

f (x ) =

f (x ) =

{

α α − 1 − xα x e β 1

σ 2π

−

e

β

( )

1 x−μ 2 σ

β1 α Γ[(α + 1) α ]

[ ( ) ( )]

β2 α Γ

α +1 α +1 − Γ2 α α

2

2(x − a ) if a ≤ x ≤ m (b − a )(m − a ) 2(b − x ) if m < x ≤ b (b − a )(b − m )

μ

σ2

a+b+m 3

a 2 + b 2 + m 2 − ab − am − bm 18


225

HYPOTHESIS TESTING Table A. Tests on means of normal distribution—variance known. Hypothesis

Test Statistic

Criteria for Rejection

H0: μ = μ0 H1: μ ≠ μ0

|Z0| > Zα /2

H0: μ = μ0 H0: μ < μ0

Z0 ≡

X − μ0 σ

Z0 < –Z α

n

H0: μ = μ0 H1: μ > μ0

Z0 > Z α

H0: μ1 – μ2 = γ H1: μ1 – μ2 ≠ γ

|Z0| > Zα /2

X1 − X 2 − γ

Z0 ≡

H0: μ1 – μ2 = γ H1: μ1 – μ2 < γ

Z0 γ

Z0 > Z α

Table B. Tests on means of normal distribution—variance unknown. Hypothesis

Test Statistic


H0: μ = μ0 H1: μ ≠ μ0

|t0| > tα /2, n – 1

H0: μ = μ0 H1: μ < μ0

t0 =

X − μ0 S

t0 < –t α , n – 1

n

H0: μ = μ0 H1: μ > μ0

t0 > tα , n – 1

t0 =

H0: μ1 – μ2 = γ H1: μ1 – μ2 ≠ γ

X1 − X 2 − γ 1 1 + n1 n2

Sp

Variances equal

|t0| > t α /2, v

v = n1 + n2 − 2 t0 =

H0: μ1 – μ2 = γ H1: μ1 – μ2 < γ

X1 − X 2 − γ S12 S22 + n1 n2

Variances unequal

H0: μ1 – μ2 = γ H1: μ1 – μ2 > γ

ν=

(

S12 S22 + n1 n2 2

(S n ) + (S 2 1

1

n1 − 1

In Table B,

226

Sp2


= [(n1 –

1)S12

+ (n2 –

1)S22]/v

t0 < –t α , v 2

) 2 2

n2

)

n2 − 1

2

t0 > tα , v

Table C. Tests on variances of normal distribution with unknown mean. Hypothesis

Test Statistic


H0: σ 2 = σ 02 H1: σ 2 ≠ σ 02

χ 02 > χ 2 α 2 , n − 1 or χ 02 < χ 21− α 2 , n −1

H0: σ 2 = σ 02 H1: σ 2 < σ 02

χ 02 =

(n − 1)S 2

χ 02 < χ 2 1– α /2, n − 1

σ 02

H0: σ 2 = σ 02 H1: σ 2 > σ 02

χ 02 > χ 2 α , n − 1

H0: σ 12 = σ 22 H1: σ 12 ≠ σ 22

F0 =

S12 S 22

F0 > Fα 2 , n1 −1, n2 −1 F0 < F1−α 2 , n1 −1, n2 −1

H0: σ 12 = σ 22 H1: σ 12 < σ 22

F0 =

S 22 S12

F0 > Fα , n2 −1, n1 −1

H0: σ 12 = σ 22 H1: σ 12 > σ 22

F0 =

S12 S 22

F0 > Fα , n1 −1, n2 −1

ANTHROPOMETRIC MEASUREMENTS

SITTING HEIGHT (ERECT)

SITTING HEIGHT (NORMAL)

THIGH CLEARANCE HEIGHT

ELBOW-TO-ELBOW BREADTH

KNEE HEIGHT ELBOW REST HEIGHT

SEAT BREADTH

BUTTOCK POPLITEAL LENGTH

BUTTOCK KNEE LENGTH

POPLITEAL HEIGHT

(AFTER SANDERS AND McCORMICK, HUMAN FACTORS IN DESIGN, McGRAW HILL, 1987)


227

U.S. Civilian Body Dimensions, Female/Male, for Ages 20 to 60 Years (Centimeters) (See Anthropometric Measurements Figure) HEIGHTS Stature (height) Eye height Shoulder (acromion) height Elbow height Knuckle height Height, sitting Eye height, sitting Shoulder height, sitting Elbow rest height, sitting Knee height, sitting Popliteal height, sitting Thigh clearance height DEPTHS Chest depth Elbow-fingertip distance Buttock-knee length, sitting Buttock-popliteal length, sitting Forward reach, functional BREADTHS Elbow-to-elbow breadth Hip breadth, sitting HEAD DIMENSIONS Head breadth Head circumference Interpupillary distance HAND DIMENSIONS Hand length Breadth, metacarpal Circumference, metacarpal Thickness, metacarpal III Digit 1 Breadth, interphalangeal Crotch-tip length Digit 2 Breadth, distal joint Crotch-tip length Digit 3 Breadth, distal joint Crotch-tip length Digit 4 Breadth, distal joint Crotch-tip length Digit 5 Breadth, distal joint Crotch-tip length FOOT DIMENSIONS Foot length Foot breadth Lateral malleolus height Weight (kg)

228


Percentiles 5th

50th

95th

Std. Dev.

149.5 / 161.8 138.3 / 151.1 121.1 / 132.3 93.6 / 100.0 64.3 / 69.8 78.6 / 84.2 67.5 / 72.6 49.2 / 52.7 18.1 / 19.0 45.2 / 49.3 35.5 / 39.2 10.6 / 11.4

160.5 / 173.6 148.9 / 162.4 131.1 / 142.8 101.2 / 109.9 70.2 / 75.4 85.0 / 90.6 73.3 / 78.6 55.7 / 59.4 23.3 / 24.3 49.8 / 54.3 39.8 / 44.2 13.7 / 14.4

171.3 / 184.4 159.3 / 172.7 141.9 / 152.4 108.8 / 119.0 75.9 / 80.4 90.7 / 96.7 78.5 / 84.4 61.7 / 65.8 28.1 / 29.4 54.5 / 59.3 44.3 / 48.8 17.5 / 17.7

6.6 / 6.9 6.4 / 6.6 6.1 / 6.1 4.6 / 5.8 3.5 / 3.2 3.5 / 3.7 3.3 / 3.6 3.8 / 4.0 2.9 / 3.0 2.7 / 2.9 2.6 / 2.8 1.8 / 1.7

21.4 / 21.4 38.5 / 44.1 51.8 / 54.0 43.0 / 44.2 64.0 / 76.3

24.2 / 24.2 42.1 / 47.9 56.9 / 59.4 48.1 / 49.5 71.0 / 82.5

29.7 / 27.6 46.0 / 51.4 62.5 / 64.2 53.5 / 54.8 79.0 / 88.3

2.5 / 1.9 2.2 / 2.2 3.1 / 3.0 3.1 / 3.0 4.5 / 5.0

31.5 / 35.0 31.2 / 30.8

38.4 / 41.7 36.4 / 35.4

49.1 / 50.6 43.7 / 40.6

5.4 / 4.6 3.7 / 2.8

13.6 / 14.4 52.3 / 53.8 5.1 / 5.5

14.54 / 15.42 54.9 / 56.8 5.83 / 6.20

15.5 / 16.4 57.7 / 59.3 6.5 / 6.8

0.57 / 0.59 1.63 / 1.68 0.4 / 0.39

16.4 / 17.6 7.0 / 8.2 16.9 / 19.9 2.5 / 2.4

17.95 / 19.05 7.66 / 8.88 18.36 / 21.55 2.77 / 2.76

19.8 / 20.6 8.4 / 9.8 19.9 / 23.5 3.1 / 3.1

1.04 / 0.93 0.41 / 0.47 0.89 / 1.09 0.18 / 0.21

1.7 / 2.1 4.7 / 5.1

1.98 / 2.29 5.36 / 5.88

2.1 / 2.5 6.1 / 6.6

0.12 / 0.13 0.44 / 0.45

1.4 / 1.7 6.1 / 6.8

1.55 / 1.85 6.88 / 7.52

1.7 / 2.0 7.8 / 8.2

0.10 / 0.12 0.52 / 0.46

1.4 / 1.7 7.0 / 7.8

1.53 / 1.85 7.77 / 8.53

1.7 / 2.0 8.7 / 9.5

0.09 / 0.12 0.51 / 0.51

1.3 / 1.6 6.5 / 7.4

1.42 / 1.70 7.29 / 7.99

1.6 / 1.9 8.2 / 8.9

0.09 / 0.11 0.53 / 0.47

1.2 / 1.4 4.8 / 5.4

1.32 / 1.57 5.44 / 6.08

1.5 / 1.8 6.2 / 6.99

0.09/0.12 0.44/0.47

22.3 / 24.8 8.1 / 9.0 5.8 / 6.2

24.1 / 26.9 8.84 / 9.79 6.78 / 7.03

26.2 / 29.0 9.7 / 10.7 7.8 / 8.0

1.19 / 1.28 0.50 / 0.53 0.59 / 0.54

46.2 / 56.2

61.1 / 74.0

89.9 / 97.1

13.8 / 12.6

ERGONOMICS—HEARING The average shifts with age of the threshold of hearing for pure tones of persons with “normal” hearing, using a 25-year-old group as a reference group.

-

Equivalent sound-level contours used in determining the A-weighted sound level on the basis of an octave-band analysis. The curve at the point of the highest penetration of the noise spectrum reflects the A-weighted sound level.


229

Estimated average trend curves for net hearing loss at 1,000, 2,000, and 4,000 Hz after continuous exposure to steady noise. Data are corrected for age, but not for temporary threshold shift. Dotted portions of curves represent extrapolation from available data.

Tentative upper limit of effective temperature (ET) for unimpaired mental performance as related to exposure time; data are based on an analysis of 15 studies. Comparative curves of tolerable and marginal physiological limits are also given.

Effective temperature (ET) is the dry bulb temperature at 50% relative humidity, which results in the same physiological effect as the present conditions.

230


MECHANICAL ENGINEERING MECHANICAL DESIGN AND ANALYSIS Stress Analysis See MECHANICS OF MATERIALS section. Failure Theories See MECHANICS OF MATERIALS section and the MATERIALS SCIENCE section. Deformation and Stiffness See MECHANICS OF MATERIALS section. Components Square Thread Power Screws: The torque required to raise, TR, or to lower, TL, a load is given by TR =

Fdm l + rndm Fncdc , d n 2 rdm - nl + 2

TL =

Fdm rndm - l Fncdc , where d n 2 rdm + nl + 2

dc = mean collar diameter, dm = mean thread diameter, l = lead, F = load, μ = coefficient of friction for thread, and μc = coefficient of friction for collar. The efficiency of a power screw may be expressed as η = Fl/(2πT)

Mechanical Springs Helical Linear Springs: The shear stress in a helical linear spring is

Spring Material: The minimum tensile strength of common spring steels may be determined from Sut = A/d m where Sut is the tensile strength in MPa, d is the wire diameter in millimeters, and A and m are listed in the following table: Material Music wire Oil-tempered wire Hard-drawn wire Chrome vanadium Chrome silicon

The deflection and force are related by F = kx where the spring rate (spring constant) k is given by 4 k = d 3G 8D N

where G is the shear modulus of elasticity and N is the number of active coils. See Table of Material Properties at the end of the MECHANICS OF MATERIALS section for values of G.

A 2060 1610 1510 1790 1960

Maximum allowable torsional stress for static applications may be approximated as Ssy = τ = 0.45Sut cold-drawn carbon steel (A227, A228, A229) Ssy = τ = 0.50Sut hardened and tempered carbon and low-alloy steels (A232, A401)

Compression Spring Dimensions Type of Spring Ends Term End coils, Ne Total coils, Nt Free length, L0 Solid length, Ls Pitch, p

, where x = Ks 8FD rd 3 d = wire diameter, F = applied force, D = mean spring diameter Ks = (2C + 1)/(2C), and C = D/d.

m 0.163 0.193 0.201 0.155 0.091

ASTM A228 A229 A227 A232 A401

Term End coils, Ne Total coils, Nt Free length, L0 Solid length, Ls Pitch, p

Plain and Ground

Plain 0 N pN + d d (Nt + 1) (L0 – d)/N

1 N+1 p (N + 1) dNt L0/(N + 1)

Squared or Closed 2 N+2 pN + 3d d (Nt + 1) (L0 – 3d)/N

Squared and Ground 2 N+2 pN + 2d dNt (L0 – 2d)/N

Helical Torsion Springs: The bending stress is given as σ = Ki [32Fr/(πd 3 )] where F is the applied load and r is the radius from the center of the coil to the load. Ki = correction factor = (4C2 – C – 1) / [4C (C – 1)] C = D/d

MECHANICAL ENGINEERING

231

The deflection θ and moment Fr are related by Fr = kθ where the spring rate k is given by 4 k= dE 64DN

where k has units of N•m/rad and θ is in radians.

Spring Material: The strength of the spring wire may be found as shown in the section on linear springs. The allowable stress σ is then given by Sy = σ = 0.78Sut cold-drawn carbon steel (A227, A228, A229)

Intermediate- and Long-Length Columns The slenderness ratio of a column is Sr = l/k, where l is the length of the column and k is the radius of gyration. The radius of gyration of a column cross-section is, k = I A where I is the area moment of inertia and A is the cross-sectional area of the column. A column is considered to be intermediate if its slenderness ratio is less than or equal to (Sr)D, where

^Sr hD = r 2SE , and y

E = Young’s modulus of respective member, and Sy = yield strength of the column material.

For intermediate columns, the critical load is

Sy = σ = 0.87Sut hardened and tempered carbon and low-alloy steel (A232, A401)

Ball/Roller Bearing Selection The minimum required basic load rating (load for which 90% of the bearings from a given population will survive 1 million revolutions) is given by C = PL1 a, where C P L a

= minimum required basic load rating, = design radial load, = design life (in millions of revolutions), and = 3 for ball bearings, 10/3 for roller bearings.

When a ball bearing is subjected to both radial and axial loads, an equivalent radial load must be used in the equation above. The equivalent radial load is Peq = XVFr + YFa, where Peq = equivalent radial load, Fr = applied constant radial load, and Fa = applied constant axial (thrust) load. For radial contact, deep-groove ball bearings: V = 1 if inner ring rotating, 1.2 if outer ring rotating, If Fa /(VFr) > e, - 0.247

F X = 0.56, and Y = 0.840 d a n C 0

F where e = 0.513 d a n C

0.236

, and

0

C0 = basic static load rating from bearing catalog. If Fa /(VFr) ≤ e, X = 1 and Y = 0.

232


SySr H n , where Pcr = A > Sy - 1 d E 2r 2

Pcr = critical buckling load, A = cross-sectional area of the column, Sy = yield strength of the column material, E = Young’s modulus of respective member, and Sr = slenderness ratio.

For long columns, the critical load is 2 Pcr = r EA Sr2

where the variables are as defined above. For both intermediate and long columns, the effective column length depends on the end conditions. The AISC recommended values for the effective lengths of columns are, for: rounded-rounded or pinned-pinned ends, leff = l; fixed-free, leff = 2.1l; fixed-pinned, leff = 0.80l; fixed-fixed, leff = 0.65l. The effective column length should be used when calculating the slenderness ratio. Power Transmission Shafts and Axles Static Loading: The maximum shear stress and the von Mises stress may be calculated in terms of the loads from 1 2 xmax = 2 3 8^8M + Fd h2 + ^8T h2B , rd 1 2 vl = 4 3 8^8M + Fd h2 + 48T 2B , where rd

M F T d

= the bending moment, = the axial load, = the torque, and = the diameter.

Fatigue Loading: Using the maximum-shear-stress theory combined with the Soderberg line for fatigue, the diameter and safety factor are related by R V1 2 2 2 rd3 = n Sf Mm + K f Ma p + f Tm + K fsTa p W SS S 32 Se Sy Se WW y T X where d = diameter, n = safety factor, Ma = alternating moment, Mm = mean moment, Ta = alternating torque, Tm = mean torque, Se = fatigue limit, Sy = yield strength, Kf = fatigue strength reduction factor, and Kfs = fatigue strength reduction factor for shear. Joining Threaded Fasteners: The load carried by a bolt in a threaded connection is given by Fb = CP + Fi

Fm < 0

while the load carried by the members is Fm = (1 – C) P – Fi

Coefficients A and b are given in the table below for various joint member materials. Material Steel Aluminum Copper Gray cast iron

A 0.78715 0.79670 0.79568 0.77871

b 0.62873 0.63816 0.63553 0.61616

The approximate tightening torque required for a given preload Fi and for a steel bolt in a steel member is given by T = 0.2 Fid.

Threaded Fasteners – Design Factors: The bolt load factor is nb = (SpAt – Fi)/CP The factor of safety guarding against joint separation is ns = Fi / [P (1 – C)]

Threaded Fasteners – Fatigue Loading: If the externally applied load varies between zero and P, the alternating stress is σa = CP/(2At) and the mean stress is σm = σa + Fi /At

Bolted and Riveted Joints Loaded in Shear:

Fm < 0, where

C = joint coefficient, = kb/(kb + km) Fb = total bolt load, Fi = bolt preload, Fm = total material load, P = externally applied load, kb = the effective stiffness of the bolt or fastener in the grip, and km = the effective stiffness of the members in the grip.

Failure by pure shear, (a) τ = F/A, where F = shear load, and A = cross-sectional area of bolt or rivet.

Bolt stiffness may be calculated from kb =

Ad AtE , where Ad lt + Atld

Ad = major-diameter area, At = tensile-stress area, E = modulus of elasticity, ld = length of unthreaded shank, and lt = length of threaded shank contained within the grip.

Failure by rupture, (b) σ = F/A, where F = load and A = net cross-sectional area of thinnest member.

If all members within the grip are of the same material, member stiffness may be obtained from km = dEAeb(d/l), where d = bolt diameter, E = modulus of elasticity of members, and l = grip length. MECHANICAL ENGINEERING

233

Failure by crushing of rivet or member, (c) σ = F/A, where F = load and A = projected area of a single rivet.

Press/Shrink Fits The interface pressure induced by a press/shrink fit is p=

0.5d 2 2 2 2 r r + r fo r f r + ri + v p v p + + o i Eo ro2 - r 2 Ei r 2 - ri2

where the subscripts i and o stand for the inner and outer member, respectively, and p = inside pressure on the outer member and outside pressure on the inner member, δ = the diametral interference, r = nominal interference radius, ri = inside radius of inner member, ro = outside radius of outer member, E = Young’s modulus of respective member, and v = Poisson’s ratio of respective member.

Fastener groups in shear, (d)

See the MECHANICS OF MATERIALS section on thick-wall cylinders for the stresses at the interface.

The location of the centroid of a fastener group with respect to any convenient coordinate frame is:

The maximum torque that can be transmitted by a press fit joint is approximately

x=

i=1 n

! Ai

T = 2πr2μpl,

n

n

! Ai xi

! Ai yi

, y=

i=1

i=1 n

, where

! Ai

i=1

n = total number of fasteners, i = the index number of a particular fastener, Ai = cross-sectional area of the ith fastener, xi = x-coordinate of the center of the ith fastener, and yi = y-coordinate of the center of the ith fastener. The total shear force on a fastener is the vector sum of the force due to direct shear P and the force due to the moment M acting on the group at its centroid. The magnitude of the direct shear force due to P is F1i = P n. This force acts in the same direction as P. The magnitude of the shear force due to M is F2i =

Mri n

! ri2

.

i=1

This force acts perpendicular to a line drawn from the group centroid to the center of a particular fastener. Its sense is such that its moment is in the same direction (CW or CCW) as M.

234


where r and p are defined above, T = torque capacity of the joint, μ = coefficient of friction at the interface, and l = length of hub engagement.

MANUFACTURABILITY Limits and Fits The designer is free to adopt any geometry of fit for shafts and holes that will ensure intended function. Over time, sufficient experience with common situations has resulted in the development of a standard. The metric version of the standard is newer and will be presented. The standard specifies that uppercase letters always refer to the hole, while lowercase letters always refer to the shaft.

δu

Clearance

δl

d min

D or d D max

Definitions Basic Size or nominal size, D or d, is the size to which the limits or deviations are applied. It is the same for both components. Deviation is the algebraic difference between the actual size and the corresponding basic size. Upper Deviation, δu, is the algebraic difference between the maximum limit and the corresponding basic size. Lower Deviation, δl, is the algebraic difference between the minimum limit and the corresponding basic size. Fundamental Deviation, δF, is the upper or lower deviation, depending on which is smaller. Tolerance, ΔD or Δd, is the difference between the maximum and minimum size limits of a part.

Free running fit: not used where accuracy is essential but good for large temperature variations, high running speeds, or heavy journal loads.

H9/d9

H7/g6 Sliding fit: where parts are not intended to run freely but must move and turn freely and locate accurately.

d max

Δd

ΔD

Some Preferred Fits

Locational fit: provides snug fit for location of stationary parts but can be freely assembled and disassembled.

H7/h6

Transition

Locational transition fit: for accurate location, a compromise between clearance and interference.

H7/k6

Interference

Location interference H7/p6 fit: for parts requiring rigidity and alignment with prime accuracy of location but without special bore pressure requirements. Medium drive fit: for ordinary steel parts or shrink fits on light sections. The tightest fit usable on cast iron.

H7/s6

Force fit: suitable H7/u6 for parts that can be highly stressed or for shrink fits where heavy pressing forces are impractical.

For the hole Dmax = D + DD Dmin = D For a shaft with clearance fits d, g, or h dmax = d + dF dmin = dmax - Dd

International tolerance (IT) grade numbers designate groups of tolerances such that the tolerance for a particular IT number will have the same relative accuracy for a basic size. Hole basis represents a system of fits corresponding to a basic hole size. The fundamental deviation is H. MECHANICAL ENGINEERING

235

For a shaft with transition or interference fits k, p, s, or u dmin = d + δF

Maximum Material Condition (MMC) The maximum material condition defines the dimension of a part such that the part weighs the most. The MMC of a shaft is at the maximum size of the tolerance while the MMC of a hole is at the minimum size of the tolerance.

dmax = dmin + Δd where D = d = δu = δl = δF = ΔD = Δd =

basic size of hole basic size of shaft upper deviation lower deviation fundamental deviation tolerance grade for hole tolerance grade for shaft

Least Material Condition (LMC) The least material condition or minimum material condition defines the dimensions of a part such that the part weighs the least. The LMC of a shaft is the minimum size of the tolerance while the LMC of a hole is at the maximum size of the tolerance.

The shaft tolerance is defined as Dd = du - dl

KINEMATICS, DYNAMICS, AND VIBRATIONS

International Tolerance (IT) Grades

Kinematics of Mechanisms Four-bar Linkage

Lower limit < Basic Size ≤ Upper Limit All values in mm Tolerance Grade, (∆D or ∆d)

Basic Size

IT6

IT7

IT9

0–3

0.006

0.010

0.025

3–6

0.008

0.012

0.030

6–10

0.009

0.015

0.036

10–18

0.011

0.018

0.043

18–30

0.013

0.021

0.052

30–50

0.016

0.025

0.062

Source: Preferred Metric Limits and Fits, ANSI B4.2-1978

Deviations for shafts Lower limit < Basic Size ≤ Upper Limit All values in mm Basic Size

Upper Deviation Letter, (δu) d

g

Lower Deviation Letter, (δl) h

k

p

s

u

0–3

–0.020 –0.002

0

0

+0.006

+0.014

+0.018

3–6

–0.030 –0.004

0

+0.001

+0.012

+0.019

+0.023

6–10

–0.040 –0.005

0

+0.001

+0.015

+0.023

+0.028

10–14

–0.095 –0.006

0

+0.001

+0.018

+0.028

+0.033

14–18

–0.050 –0.006

0

+0.001

+0.018

+0.028

+0.033

18–24

–0.065 –0.007

0

+0.002

+0.022

+0.035

+0.041

24–30

–0.065 –0.007

0

+0.002

+0.022

+0.035

+0.048

30–40

–0.080 –0.009

0

+0.002

+0.026

+0.043

+0.060

40–50

–0.080 –0.009

0

+0.002

+0.026

+0.043

+0.070

Source: Preferred Metric Limits and Fits, ANSI B4.2-1978

As an example, 34H7/s6 denotes a basic size of D = d = 34 mm , an IT class of 7 for the hole, and an IT class of 6 and an “s” fit class for the shaft.

The four-bar linkage shown above consists of a reference (usually grounded) link (1), a crank (input) link (2), a coupler link (3), and an output link (4). Links 2 and 4 rotate about the fixed pivots O2 and O4, respectively. Link 3 is joined to link 2 at the moving pivot A and to link 4 at the moving pivot B. The lengths of links 2, 3, 4, and 1 are a, b, c, and d, respectively. Taking link 1 (ground) as the reference (X-axis), the angles that links 2, 3, and 4 make with the axis are θ2, θ3, and θ4, respectively. It is possible to assemble a four-bar in two different configurations for a given position of the input link (2). These are known as the “open” and “crossed” positions or circuits. Position Analysis. Given a, b, c, and d, and θ2 i41, 2 = 2 arctan d -


B 2 - 4AC n 2A

where A = cos θ2 – K1 – K2 cos θ2 + K3 B = – 2sin θ2 C = K1 – (K2 + 1) cos θ2 + K3, and 2 2 2 2 K1 = da , K2 = dc , K3 = a - b + c + d 2ac

In the equation for θ4, using the minus sign in front of the radical yields the open solution. Using the plus sign yields the crossed solution. i31, 2 = 2 arctan d -

236

B!

E!

E 2 - 4DF n 2D

where D = cos θ2 – K1 + K4 cos θ2 + K5 E = –2sin θ2 F = K1 + (K4 – 1) cos θ2 + K5, and 2 2 2 2 K4 = d , K5 = c - d - a - b b 2ab

In the equation for θ3, using the minus sign in front of the radical yields the open solution. Using the plus sign yields the crossed solution. Velocity Analysis. Given a, b, c, and d, θ2, θ3, θ4, and ω2 ~3 =

a~2 sin _i4 - i2i b sin _i3 - i4i

a~ sin _i2 - i3i ~4 = c 2 sin _i4 - i3i

VAx =-a~2 sin i2,

VAy = a~2 cos i2

VBAx =-b~3 sin i3,

VBAy = b~3 cos i3

VBx =-c~4 sin i4,

VBy = c~4 cos i4

See also Instantaneous Centers of Rotation in the DYNAMICS section. Acceleration analysis. Given a, b, c, and d, θ2, θ3, θ4, and ω2, ω3, ω4, and α2 a3 = CD - AF , a4 = CE - BF , where AE - BD AE - BD A = csinθ4, B = bsinθ3 C = aa2 sin i2 + a~ 22 cos i2 + b~32 cos i3 - c~ 24 cos i4 D = ccosθ4, E = bcosθ3 F = aa2cos i2 - a~ 22sin i2 - b~32sin i3 + c~ 24sin i4 Gearing Involute Gear Tooth Nomenclature Circular pitch pc = πd/N Base pitch pb = pc cos φ Module m = d/N Center distance C = (d1 + d2)/2 where N = number of teeth on pinion or gear d = pitch circle diameter φ = pressure angle

Gear Trains: Velocity ratio, mv, is the ratio of the output velocity to the input velocity. Thus, mv = ωout /ωin. For a two-gear train, mv = –Nin /Nout where Nin is the number of teeth on the input gear and Nout is the number of teeth on the output gear. The negative sign indicates that the output gear rotates in the opposite sense with respect to the input gear. In a compound gear train, at least one shaft carries more than one gear (rotating at the same speed). The velocity ratio for a compound train is:

mv = !

product of number of teeth on driver gears product of number of teeth on driven gears

A simple planetary gearset has a sun gear, an arm that rotates about the sun gear axis, one or more gears (planets) that rotate about a point on the arm, and a ring (internal) gear that is concentric with the sun gear. The planet gear(s) mesh with the sun gear on one side and with the ring gear on the other. A planetary gearset has two independent inputs and one output (or two outputs and one input, as in a differential gearset). Often one of the inputs is zero, which is achieved by grounding either the sun or the ring gear. The velocities in a planetary set are related by ~L - ~arm ~ f - ~arm = ! mv, where ωf = speed of the first gear in the train, ωL = speed of the last gear in the train, and ωarm = speed of the arm. Neither the first nor the last gear can be one that has planetary motion. In determining mv, it is helpful to invert the mechanism by grounding the arm and releasing any gears that are grounded. Dynamics of Mechanisms Gearing Loading on Straight Spur Gears: The load, W, on straight spur gears is transmitted along a plane that, in edge view, is called the line of action. This line makes an angle with a tangent line to the pitch circle that is called the pressure angle φ. Thus, the contact force has two components: one in the tangential direction, Wt, and one in the radial direction, Wr. These components are related to the pressure angle by Wr = Wt tan(φ). Only the tangential component Wt transmits torque from one gear to another. Neglecting friction, the transmitted force may be found if either the transmitted torque or power is known: Wt = 2T = 2T , mN d 2 H 2H , where Wt = = mN ~ d~ Wt = transmitted force (newton), T = torque on the gear (newton-mm), d = pitch diameter of the gear (mm), N = number of teeth on the gear, m = gear module (mm) (same for both gears in mesh), H = power (kW), and ω = speed of gear (rad/sec).


237

Stresses in Spur Gears: Spur gears can fail in either bending (as a cantilever beam, near the root) or by surface fatigue due to contact stresses near the pitch circle. AGMA Standard 2001 gives equations for bending stress and surface stress. They are: vb =

Dynamic (Two-plane) Balance A

Wt KaKm K K K , bending and FmJ Kv S B I Wt CaCm C C , surface stress, where FId Cv s f

vc = Cp

σb = bending stress, σc = surface stress, Wt = transmitted load, F = face width, m = module, J = bending strength geometry factor, Ka = application factor, KB = rim thickness factor, KI = idler factor, Km = load distribution factor, Ks = size factor, Kv = dynamic factor, Cp = elastic coefficient, I = surface geometry factor, d = pitch diameter of gear being analyzed, and Cf = surface finish factor. Ca, Cm, Cs, and Cv are the same as Ka, Km, Ks, and Kv, respectively.

Rigid Body Dynamics See DYNAMICS section. Natural Frequency and Resonance See DYNAMICS section.

B

Two balance masses are added (or subtracted), one each on planes A and B. n n mBRBx =- 1 ! miRixli, mBRBy =- 1 ! miRiyli lB i = 1 lB i = 1 n

mARAx =- ! miRix - mBRBx i=1 n

mARAy =- ! miRiy - mBRBy i=1

where mA = balance mass in the A plane mB = balance mass in the B plane RA = radial distance to CG of balance mass RB = radial distance to CG of balance mass and θA, θB, RA, and RB are found using the relationships given in Static Balance above. Balancing Equipment The figure below shows a schematic representation of a tire/wheel balancing machine.

Balancing of Rotating and Reciprocating Equipment Static (Single-plane) Balance n

n

i=1

i=1

mbRbx =- ! miRix, mbRby =- ! miRiy ib = arctan e

mbRby o mbRbx

mbRb = _ mbRbxi + _mbRby i 2

where mb Rb mi Ri θb

2

= balance mass = radial distance to CG of balance mass = ith point mass = radial distance to CG of the ith point mass = angle of rotation of balance mass CG with respect to a reference axis x, y = subscripts that designate orthogonal components

238


Ignoring the weight of the tire and its reactions at 1 and 2, F1x + F2x + mARAxω2 + mBRBxω2 = 0 F1y + F2y + mARAyω2 + mBRByω2 = 0 F1xl1 + F2xl2 + mBRBxω2lB = 0 F1yl1 + F2yl2 + mBRByω2 lB = 0

F1xl1 + F2xl2 lB ~ 2 F1yl1 + F2yl2 mBRBy = lB ~ 2 F +F mARAx =- 1x 2 2x - mbRBx ~ F1y + F2y mARAy =- mbRBy ~2 mBRBx =

MATERIALS AND PROCESSING Mechanical and Thermal Properties See MATERIALS SCIENCE section. Thermal Processing See MATERIALS SCIENCE section. Testing See MECHANICS OF MATERIALS section.

THERMODYNAMICS AND ENERGY CONVERSION PROCESSES Ideal and Real Gases See THERMODYNAMICS section. Reversibility/Irreversibility See THERMODYNAMICS section. Thermodynamic Equilibrium See THERMODYNAMICS section. Psychrometrics See additional material in THERMODYNAMICS section. HVAC—Pure Heating and Cooling MOIST AIR

ma

2

1 Q

MEASUREMENTS, INSTRUMENTATION, AND CONTROL Mathematical Fundamentals See DIFFERENTIAL EQUATIONS and LAPLACE TRANSFORMS in the MATHEMATICS section, and CONTROL SYSTEMS in the MEASUREMENT and CONTROLS section. System Descriptions See LAPLACE TRANSFORMS in the ELECTRICAL and COMPUTER ENGINEERING section, and CONTROL SYSTEMS in the MEASUREMENT and CONTROLS section. Sensors and Signal Conditioning See the Measurements segment of the MEASUREMENT and CONTROLS section and the Analog Filter Circuits segment of the ELECTRICAL and COMPUTER ENGINEERING section. Data Collection and Processing See the Sampling segment of the MEASUREMENT and CONTROLS section. Dynamic Response See CONTROL SYSTEMS in the MEASUREMENT and CONTROLS section.

1

2

or 2

1 T

Qo = mo a ^h2 - h1h = mo acpm ^T2 - T1h cpm = 1.02 kJ _ kg : cCi

Cooling and Dehumidification Qout MOIST AIR ma 1

2 3

LIQUID OUT mw


239

ω

ω 1 2

T

2

wb

=C

ons

tan

t

1 T

T

Qo out = mo a 8^h1 - h2h - h f3 ^~1 - ~2hB

h2 = h1 + h3 ^~2 - ~1h

mo w = mo a ^~1 - ~2h

mo w = mo a ^~2 - ~1h h3 = h f at Twb

Heating and Humidification

Adiabatic Mixing

3 Qin

MOIST AIR

WATER OR STEAM

MOIST AIR

ma1

3

1

ma 2

1

2

4, 4'

MOIST AIR ma2

ω

ω

2

4

4'

3 1

1

2 T

Qo in = mo a ^h2 - h1h

T

mo w = mo a ^~4 l - ~2h or mo w = ^~4 - ~2h

mo a3 = mo a1 + mo a2 mo h + mo a2h2 h3 = a1 1 mo a3 mo ~ + mo a2 ~2 ~3 = a1 1 mo a3

Adiabatic Humidification (evaporative cooling) MOIST AIR

1

WATER 3

240

mo a2 # distance 12 measured on mo a3 psychrometric chart

distance 13 =

ma


2

ma3

Performance of Components Fans, Pumps, and Compressors Scaling Laws

d

Q n ND3 2

=d

d

mo mo n =d n tND3 2 tND3 1

d

H H n = d 2 2n N 2D 2 2 ND 1

d

P P n = d 2 2n tN 2 D 2 2 tN D 1

e

Wo Wo o 3 5o = e tN D 2 tN3D5 1

Centrifugal Pump Characteristics PUMP PERFORMANCE CURVES (CONSTANT N, D, ρ)

Q n ND3 1

η

H

POWER

where Q = volumetric flow rate, mo = mass flow rate, H = head, P = pressure rise, Wo = power, ρ = fluid density, N = rotational speed, and D = impeller diameter.

NPSH REQUIRED FLOW RATE, Q

Net Positive Suction Head (NPSH) V2 P P NPSH = tgi + i - tgv , where 2g

Subscripts 1 and 2 refer to different but similar machines or to different operating conditions of the same machine. Fan Characteristics Δp

Pi = inlet pressure to pump, Vi = velocity at inlet to pump, and Pv = vapor pressure of fluid being pumped. Fluid power Wo fluid = tgHQ tgHQ Pump ^ brakeh power Wo = h pump o Purchased power Wo purchased = hW motor

hpump = pump efficiency ^0 to 1h

hmotor = motor efficiency ^0 to 1h H = head increase provided by pump

POWER

Cycles and Processes Internal Combustion Engines Otto Cycle (see THERMODYNAMICS section) Diesel Cycle CONSTANT N, D, ρ

Typical Backward Curved Fans DPQ Wo = h , where f Wo = fan power, ΔP = pressure rise, and ηf = fan efficiency.


241

Indicated Thermal Efficiency Wo i hi = mo f ^ HV h Mechanical Efficiency h Wo hi = ob = hb i Wi

r = V1 V2 rc = V3 V2 h=1-

k 1 > rc - 1 H r k - 1 k _ rc - 1i

k = cp cv Brake Power Wo b = 2rTN = 2rFRN, where Wo b T N F R

Displacement Volume = brake power (W), = torque (N•m), = rotation speed (rev/s), = force at end of brake arm (N), and = length of brake arm (m).

2 Vd = rB S , m3 for each cylinder 4

Total volume = Vt = Vd + Vc, m3 Vc = clearance volume (m3). Compression Ratio rc = Vt /Vc Mean Effective Pressure (mep) o s Wn mep = , where Vd ncN ns = number of crank revolutions per power stroke, nc = number of cylinders, and Vd = displacement volume per cylinder.

Indicated Power Wo i = Wo b + Wo f , where Wo i = indicated power (W), and Wo f = friction power (W).

mep can be based on brake power (bmep), indicated power (imep), or friction power (fmep). Volumetric Efficiency 2mo a (four-stroke cycles only) hv = taVd ncN

Brake Thermal Efficiency Wo b hb = , where mo f ^ HV h

where mo a = mass flow rate of air into engine _ kg si, and

ηb = brake thermal efficiency, mo f = fuel consumption rate (kg/s), and HV = heating value of fuel (J/kg).

Specific Fuel Consumption (SFC)

ta = density of air ` kg m3j .

mo f sfc = o = 1 , kg J hHV W Use hb and Wo b for bsfc and hiand Wo i for isfc.

242


Gas Turbines Brayton Cycle (Steady-Flow Cycle)

Steam Power Plants Feedwater Heaters 2

m2 m1

m1 + m2 OPEN 3

1

m1 h1 + m2 h2 = h3 ( m1 + m2 )

2

m2 m1

m1 CLOSED

3

4

1

m2

m1 h1 + m2 h2 = m1 h3 + m2 h4

Steam Trap

LIQUID + VAPOR

2 1

m

LIQUID ONLY

w12 = h1 – h2 = cp (T1 – T2) w34 = h3 – h4 = cp (T3 – T4) wnet = w12 + w34 q23 = h3 – h2 = cp (T3 – T2) q41 = h1 – h4 = cp (T1 – T4) qnet = q23 + q41 η = wnet /q23

h 2 = h1


243

The thermal storage tank is sized to defer most or all of the chilled water requirements during the electric utility’s peak demand period, thus reducing electrical demand charges. The figure above shows a utility demand window of 8 hours (noon to 8 pm), but the actual on-peak period will vary from utility to utility. The Monthly Demand Reduction (MDR), in dollars per month, is

Junction m1 + m2

m1

3

1

2

m2

MDR = ΔPon-peak R, where m1 h1 + m2 h2 = h3 ( m1 + m2 )

ΔPon-peak = Reduced on-peak power, kW R = On-peak demand rate, $/kW/month The MDR is also the difference between the demand charge without energy storage and that when energy storage is in operation.

Pump ηp

A typical utility rate structure might be four months of peak demand rates (June – September) and eight months of off-peak demand rates (October – May). The customer’s utility obligation will be the sum of the demand charge and the kWh energy charge.

2 1

FLUID MECHANICS AND FLUID MACHINERY

w

Fluid Statics See FLUID MECHANICS section.

w = h 1 – h 2 = ( h 1 – h 2S) / ηP h 2S – h 1 = v ( P 2 – P 1 ) w=–

Incompressible Flow See FLUID MECHANICS section.

v ( P2 – P1 )

ηp

Fluid Machines (Incompressible) See FLUID MECHANICS section and Performance of Components above. Compressible Flow Mach Number The local speed of sound in an ideal gas is given by: c = kRT , where

See also THERMODYNAMICS section.

Combustion and Combustion Products See THERMODYNAMICS section. Energy Storage Energy storage comes in several forms, including chemical, electrical, mechanical, and thermal. Thermal storage can be either hot or cool storage. There are numerous applications in the HVAC industry where cool storage is utilized. The cool storage applications include both ice and chilled water storage. A typical chilled water storage system can be utilized to defer high electric demand rates, while taking advantage of cheaper off-peak power. A typical facility load profile is shown below.

DEMAND POWER, kW

WITH THERMAL STORAGE

MIDNIGHT

TIME OF DAY

244


6 PM

T ≡ absolute temperature This shows that the acoustic velocity in an ideal gas depends only on its temperature. The Mach number (Ma) is the ratio of the fluid velocity to the speed of sound. Ma ≡ Vc V ≡ mean fluid velocity

k

WITHOUT THERMAL STORAGE NOON

cp k ≡ ratio of specific heats = c v R ≡ gas constant

Isentropic Flow Relationships In an ideal gas for an isentropic process, the following relationships exist between static properties at any two points in the flow.

Δ Pon-peak

6 AM

c ≡ local speed of sound

MIDNIGHT

P2 T ^k - 1h t d 2n = d t2 n P1 = T1 1

k

The stagnation temperature, T0, at a point in the flow is related to the static temperature as follows:

The following equations relate downstream flow conditions to upstream flow conditions for a normal shock wave.

2

T0 = T + V 2 : cp

Ma2 =

The relationship between the static and stagnation properties (T0, P0, and ρ0) at any point in the flow can be expressed as a function of the Mach number as follows: T0 k - 1 : Ma 2 1 T = + 2 k

k

P0 c T0 m ^k - 1h 1 : Ma 2l ^k - 1h = b1 + k P = T 2 t0 c T0 m 1 1 2 k-1 t = T ^k - 1h = b1 + 2 : Ma l ^k - 1h Compressible flows are often accelerated or decelerated through a nozzle or diffuser. For subsonic flows, the velocity decreases as the flow cross-sectional area increases and vice versa. For supersonic flows, the velocity increases as the flow cross-sectional area increases and decreases as the flow cross-sectional area decreases. The point at which the Mach number is sonic is called the throat and its area is represented by the variable, A*. The following area ratio holds for any Mach number.

2k Ma12 - ^ k - 1h T2 8 2 + ^ k - 1h Ma12B = 2 T1 ^ k + 1h Ma12 P2 1 8 2k Ma 2 - ^ k - 1hB 1 P1 = k + 1

^ k + 1h Ma12 V1 t2 = = t1 V2 ^ k - 1h Ma12 + 2 T01 = T02

Fluid Machines (Compressible) Compressors Compressors consume power to add energy to the working fluid. This energy addition results in an increase in fluid pressure (head). INLET

R V ^k + 1h S1 + 1 ^ k - 1h Ma 2 W2 ^k - 1h A = 1 S 2 W Ma S 1 WW A* ^ h k 1 + S 2 T X where A ≡ area [length2] A* ≡ area at the sonic point (Ma = 1.0) Normal Shock Relationships A normal shock wave is a physical mechanism that slows a flow from supersonic to subsonic. It occurs over an infinitesimal distance. The flow upstream of a normal shock wave is always supersonic and the flow downstream is always subsonic as depicted in the figure.

^ k - 1h Ma12 + 2 2k Ma12 - ^ k - 1h

COMPRESSOR

Win

EXIT

For an adiabatic compressor with ΔPE = 0 and negligible ΔKE: Wo comp =-mo _ he - hi i For an ideal gas with constant specific heats: Wo comp =-mc o p _Te - Ti i Per unit mass: wcomp =-cp _Te - Ti i Compressor Isentropic Efficiency:

1

2

Ma > 1

Ma < 1

NORMAL SHOCK

w T - Ti where, hC = ws = es Te - Ti a wa ≡ actual compressor work per unit mass ws ≡ isentropic compressor work per unit mass Tes ≡ isentropic exit temperature (see THERMODYNAMICS section) For a compressor where ΔKE is included: V 2 - Vi2 n Wo comp =-mo d he - hi + e 2 =-mo dcp _Te - Ti i +

Ve2 - Vi2 n 2


245

Operating Characteristics See Performance of Components above.

Adiabatic Compression: Wo comp =

mo Pi k >d Pe n ^ k - 1h ti hc Pi

1 -1 k

- 1H

Lift/Drag See FLUID MECHANICS section.

Wo comp = fluid or gas power (W) Pi

= inlet or suction pressure (N/m2)

Pe

= exit or discharge pressure (N/m2)

k

= ratio of specific heats = cp/cv

ρi

= inlet gas density (kg/m )

ηc

= isentropic compressor efficiency

Impulse/Momentum See FLUID MECHANICS section.

HEAT TRANSFER

3

Conduction See HEAT TRANSFER and TRANSPORT PHENOMENA sections.

Isothermal Compression RTi P Wo comp = ln e (mo ) Mhc Pi

Convection See HEAT TRANSFER section.

Wo comp , Pi, Pe, and ηc as defined for adiabatic compression R = universal gas constant Ti = inlet temperature of gas (K) M = molecular weight of gas (kg/kmol)

Radiation See HEAT TRANSFER section.

Turbines Turbines produce power by extracting energy from a working fluid. The energy loss shows up as a decrease in fluid pressure (head).

Transient and Periodic Processes See HEAT TRANSFER section.

INLET • W out

TURBINE

Composite Walls and Insulation See HEAT TRANSFER section.

Heat Exchangers See HEAT TRANSFER section. Boiling and Condensation Heat Transfer See HEAT TRANSFER section.

REFRIGERATION AND HVAC Cycles Refrigeration and HVAC Two-Stage Cycle

EXIT

For an adiabatic turbine with ΔPE = 0 and negligible ΔKE: Wo turb = mo _ hi - hei

out

For an ideal gas with constant specific heats: Wo turb = mc o p _Ti - Tei

in, 1

Per unit mass: wturb = cp _Ti - Tei Compressor Isentropic Efficiency: w T - Te hT = wa = i T s i - Tes

in, 2

For a compressor where ΔKE is included: V 2 - Vi2 n Wo turb = mo d he - hi + e 2 = mo dcp _Te - Ti i +

246


Ve2 - Vi2 n 2

in

T

2 out

6 3

7 4

1

8

5

in

s

The following equations are valid if the mass flows are the same in each stage. Qo in h5 - h8 COPref = o = o h h1 + h6 - h5 Win, 1 + Win, 2 2 Qo out h2 - h3 COPHP = o = h2 - h1 + h6 - h5 Win, 1 + Wo in, 2

COPref = COPHP =

h1 - h4 ^h2 - h1h - _h3 - h4i

h2 - h3 ^h2 - h1h - _h3 - h4i

See also THERMODYNAMICS section. Heating and Cooling Loads Heating Load

Air Refrigeration Cycle out

in

Qo = A _Ti - Toi /Rm

L L L Rm = 1 + 1 + 2 + 3 + 1 , where h1 k1 k2 k3 h2

Qo = heat transfer rate, A = wall surface area, and R″ = thermal resistance. in

Overall heat transfer coefficient = U U = 1/R″ Qo = UA (Ti – To) Cooling Load Qo = UA (CLTD), where CLTD = effective temperature difference. CLTD depends on solar heating rate, wall or roof orientation, color, and time of day.


247

Infiltration Air change method tacpVnAC _Ti - Toi, where Qo = 3, 600 ρa = air density, cp = air specific heat, V = room volume, nAC = number of air changes per hour, Ti = indoor temperature, and To = outdoor temperature. Crack method Qo = 1.2CL _Ti - Toi where C = coefficient, and L = crack length. See also HEAT TRANSFER section. Psychrometric Charts See THERMODYNAMICS section. Coefficient of Performance (COP) See section above and THERMODYNAMICS section. Components See THERMODYNAMICS section and above sections.

248


INDEX A absolute dynamic viscosity 62 absorption (packed columns) 130 accelerated cost recovery system (ACRS) 115 AC circuits 195 acids and bases 100 AC machines 198 AC power 197 activated carbon adsorption 188 activated sludge 186 acyclic hydrocarbons 123 adiabatic humidification (evaporative cooling) 240 adiabatic mixing 240 adiabatic process 77 adsorption, activated carbon 188 aerobic digestion 187 aerodynamics 64 airfoil theory 64 air pollution 170 air refrigeration cycle 247 air stripping 188 alcohols 123 aldehydes 123 AM (amplitude modulation) 203 amorphous materials and glasses 107 anaerobic digestion 187 analog-to-digital conversion 110 analog filter circuits 206 analysis of variance (ANOVA) 218, 219 angle modulation 204 angular momentum or moment of momentum 56 anode reaction (oxidation) of a typical metal 104 ANOVA table 224 anthropometric measurements 227 Archimedes principle 63 area formulas 28 arithmetic progression 25 ASTM grain size 106 atmospheric dispersion modeling (Gaussian) 170 atmospheric stability under various conditions 170 atomic bonding 104 average values 195 Avogadro’s number 100 axles 232

B baghouse 174 balanced three-phase systems 197 balancing of rotating and reciprocating equipment 238 ball/roller bearing selection 232 band-pass filters 207 band-reject filters 207 basic cycles 75 batch reactor, constant T and V 126 batch reactor, general 127 Bayes theorem 41 beam-columns 151

beams 36, 150 deflection formulas 39 deflection of beams 37 shearing force and bending moment sign conventions 36 stiffness and moment carryover 142 stresses in beams 37 belt friction 50 benefit-cost analysis 115 Bernoulli equation 65 binary number system 213 binary phase diagrams 76, 108 iron-iron carbide phase diagram 108 lever rule 108 binomial distribution 42 biochemical pathways 93, 170 bioconcentration factor 176 biology 91 biomechanics of the human body 221 bioprocessing 98 biotower 187 bipolar junction transistor (BJT) 210 black body 88 blades 66 BOD 175 boilers, condensers, evaporators 75 bolt stiffness 233 bolt strength 152 bonds, value and yield 115 book value 115 Boolean algebra 213 brake power 242 brake thermal efficiency 242 Brayton cycle 243 break-even analysis 114 brittle materials 35 stress concentration of 105 Buckingham’s theorem 69 buoyancy 63

C calculus 25 capacitors and inductors 194 capillarity 62 capillary rise 62 capitalized costs 115 carboxylic acids 123 carcinogens 183 Carmen-Kozeny equation 190 cathode reactions (reduction) 104 cellular biology 91 composition data for biomass and selected organic compounds 99 primary subdivisions of biological organisms 91 stoichiometry of selected biological systems 96 center of buoyancy 63 center of pressure 63 centroids and moments of inertia 30 centroids of masses, areas, lengths, and volumes 49

249

chemical engineering 123 chemical interaction effects on toxicity 180 chemical process safety 132 chemical reaction engineering 126 chemical thermodynamics 125 chemical reaction equilibrium 125 reactive systems 125 vapor-liquid equilibrium 125 chemistry 100 circular sector 28 civil engineering 134 clarifier 186, 189, 190 Clausius’ statement of second law 76 closed-system availability 77 closed thermodynamic system 74 coefficient of contraction 68 coefficient of performance (COP) 75, 247, 248 cold working 104 columns, intermediate- and long-length 232 combustion and combustion products 244 combustion process 76 communication modulation types 203 communication theory and concepts 202 complex numbers 23, 196 definition 23, 164 Euler’s identity 23 polar coordinates 23 roots 23 complex power 197 composite materials 106 composite plane wall 84 composition data for biomass and selected organic compounds 99 compressible flow 244 compression ratio 242 compression spring dimensions 231 compressors 245 computer engineering 193 computer spreadsheets 109 concentrations of vaporized liquids 133 concrete 106 concurrent forces 50 condensation of a pure vapor 87 conduction 84 confidence interval 44 conic sections 21 constants, fundamental 19 continuity equation 63 continuous-stirred tank reactor (CSTR) 127 control systems 111 convection 84, 86, 87, 128 conversion factors 20 convolution 201 convolution integral 202 cooling and dehumidification 239 cooling load 247 corrosion 104 corrosion reactions 103 cost-benefit analysis 115 cost estimation 131

250

Coulomb-Mohr theory 35 CPM precedence relationships 169 critical path method (CPM) 223 critical value of stress intensity 105 critical values of the f distribution 47 critical values of X2 distribution 48 CSF, cancer slope factor 183 cumulative distribution functions 41 curl 24 current 193 curvature in rectangular coordinates 26 curvature of any curve 26 cycles 241, 246 cycles and processes 241 cyclone 173 cylindrical pressure vessel 33

D damped natural frequency 112 damped resonant frequency 112 Darcy’s law 159 Darcy-Weisbach equation 65 data quality objectives (DQO) for sampling and solids 178 deflection of beams 37 deflectors and blades 66 DeMorgan’s theorems 213 density 62 depreciation 115 derivative 25 derivatives and indefinite integrals 27 determinants 24 dew-point 75 difference equations 31, 200 differential calculus 25 tests for maximum, minimum, point of inflection 25 differential equations 30, 32 diffusers 74 diffusion 104, 128 digital signal processing 201 dimensional homogeneity and dimensional analysis 69 dimensionless group equation (Sherwood) 128 diodes 210 dispersion 40 displacement volume 242 distillation 128, 129 distortion-energy theory 35 divergence 24 dose-response curves 180 drag coefficient 64, 190, 191, 192 drag coefficients for spheres, disks, and cylinders 72 drag force 64 DSB (double-sideband modulation) 204 dual linear program 215 ductile materials 35 Dupuit’s formula 159 dynamic (two-plane) balance 238 dynamic response 239 dynamics 54 dynamics of mechanisms 237 dynamic viscosity 62

E earthwork formulas 165 economics 114 effective stack height 172 elastic strain energy 38 electrical engineering 193 electrochemistry 100 electrodialysis 190 electromagnetic dynamic fields 198 electrostatic fields 193 electrostatic precipitator efficiency 174 electrostatics 193 ellipse 21, 28 endurance limit 104 endurance limit for steels 35 endurance test 104 energy 56 energy line (Bernoulli equation) 65 energy storage 244 engineering economics 114 engineering strain 33 enthalpy 73, 75 entropy 77 environmental engineering 170 equilibrium constant 100 ergonomics 220, 228, 229 Ergun equation 67 ethers 123 ethics 121 Euler’s approximation 32 Euler’s formula 37 Euler’s Identity 23 evaporative cooling 240 exergy 77 exposure equations 184 exposure limits for selected compounds 181

F facultative pond 188 failure by crushing of rivet 234 by pure shear 233 by rupture 233 failure theories 35, 231 fan characteristics 241 Fanning friction factor 65 Faraday’s law 100, 193 fastener groups in shear 234 fate and transport 175 fatigue 35 fatigue loading 233 field equation 64 filters 207 filtration equations 190 fins 86 first-order control system models 112 first-order high-pass filters 206 first-order linear difference equation 31, 200 first-order linear differential equations 30 first-order low-pass filters 206

first-order reaction 126 first law of thermodynamics 74 flammability 132 flip-flops 214 flocculation 189 flocculator design 191 flow reactors 127 flow through a packed bed 67 fluid flow 64 fluid machines 244, 245 fluid measurements 67 fluid mechanics 62, 244 fluid statics 244 fluid temperature 85 FM (frequency modulation) 204 force 49 forces on submerged surfaces 63 forecasting 217 four-bar linkage 236 Fourier Fourier series 199 Fourier transform 31 Fourier transform and its inverse 202 Fourier transform pairs 202 Fourier transform theorems 203 Fourier’s law 84 fracture toughness, representative values 105 free convection 87 free vibration 60 frequency response and impulse response 203 Freundlich isotherm 188 friction 50, 57, 65 Froude number 69, 161

G gamma function 42 gas constant (ideal, universal) 19, 73, 104, 133, 189, 192, 244 gas flux 177 gas turbines 243 gauge factor 110 Gaussian model 170 gearing 237 gear trains: velocity ratio 237 geometric progression 25 geotech 134 Gibbs free energy 73, 77 glasses 107 gradient, divergence, and curl 24 grain size 106 gray body 88

H Hagen-Poiseuille equation 65 half-life 106, 178 hardenability 105 hazard assessment 180 hazard index 183 hazardous waste compatibility chart 182 Hazen-Williams equation 67, 161

251

head loss 65, 66 through clean bed 190 heat capacity tables 83 heat engines 75 heat exchangers 87 heating and cooling loads 247 heating and humidification 240 heating load 247 heats of reaction 100, 126 heat transfer 84, 90 heat transfer rate equations 84 helical torsion springs 231 Helmholtz free energy 77 Henry’s law at constant temperature 76 Hooke’s law 34 horizontal curve formulas 164 humidification 240 HVAC 239 cooling 239 performance of components 241 pure heating 239 refrigeration and HVAC 247 hydraulic diameter 65 hydraulic gradient (grade line) 65 hydraulic radius 65 hydrology 159 hyperbola 21 hypothesis testing 43, 226

I ideal-impulse sampling 205 ideal and real gases 239 ideal gas law 74 ideal gas mixtures 75 ideal transformers 198 impeller mixer 191 impulse-momentum principle 66 impulse response 203 impulse turbine 67 incineration 174 incomplete combustion 76 incompressible flow 244 increase of entropy principle 77 indicated power 242 indicated thermal efficiency 242 induced voltage 193 inductors 195 industrial chemicals 124 industrial engineering 215 inequality of Clausius 77 infiltration 248 inflation 114 influence lines for beams and trusses 142 instant centers 58 intake rates 184 integral calculus 26 integrals, indefinite 27 integration 32 internal combustion engines 241 inventory models 219

252

inverse transform method 217 involute gear tooth nomenclature 237 ionizing radiation equations 178 iron-iron carbide phase diagram 108 irreversibility 77 isentropic flow relationships 244 isentropic process 77

J jet propulsion 66 job sequencing 223

K Kelvin-Planck statement of second law 76 ketones 123 kinematics 54, 58, 236 kinematic viscosity 62, 65 kinetic energy 59 kinetics 56, 59, 175 Kirchhoff’s laws 194

L L’Hospital’s Rule 26 laminar flow 64 landfill 177 Laplace transforms 200 law of cosines 22 law of sines 22 learning curves 219 least squares 42, 218 Le Chatelier’s Principle 100 lethal doses 180 lever rule 108 lift coefficient 64 lift force 64 lime-soda softening equations 191 linear programming 215 linear regression 42, 218 line balancing 223 liquid-vapor equilibrium 125 liquid-vapor mixtures 76 live load reduction 143 load combinations 143 logarithms 22 log growth, population 177 logic operations and Boolean algebra 213 lossless transmission lines 199

M Mach number 244 magnetic fields 193 Manning’s equation 67, 161 manometers 62, 68 mass fraction 75 mass moment of inertia 58 mass transfer 90, 128 material handling 222 material properties 38 material safety data sheets (MSDS) 181 materials science 104 mathematics 21

matrices 23 maximum 25 maximum-normal-stress theory 35 maximum-shear-stress theory 35 maxterm 214 mean 40 mean effective pressure 242 measurement 110 measurement uncertainty 111 mechanical and thermal properties 239 mechanical design and analysis 231 mechanical efficiency 242 mechanical engineering 231 mechanical processing 104 mechanical springs 231 mechanics of materials 33 median 40 member fixed-end moments (magnitudes) 142 mensuration of areas and volumes 28 metric prefixes 19 microbial kinetics 175 minor losses in pipe fittings, contractions, and expansions 65 minterm 214 mode 33 modified ACRS factors 115 modified Goodman theory 35 Mohr’s circle - stress, 2D 34 molality of solutions 100 molarity of solutions 100 moment coefficient 64 moment of inertia 30, 49 moment of momentum 56 moments (couples) 49 moments of inertia, definitions 30 momentum 56 momentum, transfer 90 Monod kinetics 175 Moody (Stanton) diagram 71 moving average 217 multipath pipeline problems 67 multiple server model 217

N n-channel JFET and MOSFET 211 natural (free) convection 87 natural sampling 205 network optimization 215 Newton’s method for root extraction 31 Newton’s method of minimization 31 Newton’s second law for a particle 56 NIOSH formula 220 noise pollution 177 nomenclature in organic chemistry 123 non-annual compounding 114 noncarcinogens 183 normal distribution (Gaussian distribution) 42 normality of solutions 100 normal shock relationships 245 nozzles 74

NRCS (SCS) rainfall-runoff 159 number systems and codes 213 numerical integration 32 numerical methods 31 numerical integration 32 Nyquist frequency 110

O one-dimensional flows 63 opaque body 88 open-channel flow 67, 161 open-system availability 77 open thermodynamic system 74 operational amplifiers 209 organic carbon partition coefficient 176 organic chemistry, nomenclature 123 organic compounds 102 orifices 68, 69 overburden pressure 177 oxidation 100, 104

P P-h diagram for refrigerant 81 packed bed flow 67 PAM (pulse-amplitude modulation) 205 parabola 21, 28 paraboloid of revolution 29 parallel axis theorem 30, 50, 58 parallelogram 28 parallel resonance 196 Parseval’s theorem 203 partial derivative 25 partial pressures 75 partial volumes 75 partition coefficients 176 PCM (pulse-code modulation) 205 percent elongation 33 percent reduction in area (RA) 33 periodic table of elements 101 permissible noise exposures 222 permutations and combinations 40 PERT 223 pH 100 phase relations 76 photosynthesis 95 pipe flow 67 pitot tube 67 planar motion 54 plane circular motion 54 plane motion of a rigid body 58 plane truss 50 plug-flow reactor (PFR) 127 PM (phase modulation) 204 point of inflection 25 Poisson’s ratio 33 polar coordinates 23 polymer additives 107 polymers 107 population modeling 177

253

population projection equations 177 pore pressure 140 Portland cement concrete 106 power 66 power transmission 232 prefixes, metric 19 press/shrink fits 234 pressure 62, 63, 65 pressure field in a static liquid 62 primary clarifier efficiency 189 prime implicant 214 principal stresses 34 probability 40 probability and density functions: means and variances 225 probability laws 40 process capability 216 product of inertia 50 progressions and series 25 properties of water charts 70 psychrometric chart 82 psychrometrics 75 performance of components 241 pump power equation 66

Q quadratic equation 21 quadric surface (sphere) 22 quenching 104 queueing models 216

R radiation 84, 88, 178 radius of curvature 26 radius of gyration 50, 58 randomized block design 218 random variables, sums 41 random variate generation 217 Raoult’s law for vapor-liquid equilibrium 76 rapid mix and flocculator design 191 rate-of-return 115 RC and RL transients 196 reaction order 126 reactive systems 125 reference dose 183 refrigeration and HVAC 246 refrigeration cycles 75 regular polygon (n equal sides) 29 reinforced concrete design 144 relative humidity 75 reliability 219 resistivity 194 resistors in series and parallel 194 resolution of a force 49 resonance 196 resultant (two dimensions) 49 retardation factor 176 reverse osmosis 192 Reynolds number 65 rigid body dynamics 238

254

risk assessment 180 RMS values 195, 199 root locus 113 Rose equation, filtration 190 rotating and reciprocating 238 Routh test 112

S safe human dose 183 salt flux through membrane 192 sample correlation coefficient 43, 218 sampled messages 204 sampling 110 sampling and monitoring 178 scaling laws 241 SCS (NRCS) rainfall-runoff 159 second-order control system models 112 second-order linear difference equation 31, 201 second-order linear homogeneous differential equations with constant coefficients 30 second law of thermodynamics 76 sedimentation basin 189 series, mathematical 25 series resonance 196 settling equations 192 shafts and axles 232 shape factor 89 shear modulus 33 shear stress-strain 33 Sherwood number 128 signal conditioning 111 similitude 69 simple planetary gearset 237 Simpson’s 1/3 rule 165 Simpson’s rule 32 simulation 217 single-component systems 73 single server models 217 sludge age 187 sludge volume index, SVI 186 Soderberg theory 35 soil-water partition coefficient 176 soil landfill cover water balance 177 solid-state electronics and devices 209 solid/fluid separations 130 solids residence time 186 solubility product 100 source equivalents 194 specific energy diagram 161 specific gravity 62 specific volume 62 specific weight 62 sphere 28 spreadsheets 109 spring constant 231 springs 231 spur gears 238 square thread power screws 231 SSB (single-sideband modulation) 204

stack height 170 standard deviation charts 216 standard error of estimate 43, 218 standard tensile test 104 standard time determination 222 state-variable control system models 113 state functions (properties) 73 statically determinate truss 50 static balance 238 static loading failure theories 35 statics 49 statistical quality control 215 statistics 40 steady-state gain 112 steady-state reactor parameters 176 steady-state systems 74 steam power plants 243 steam tables 79 steel structures 150 stoichiometry of selected biological systems 96 Stokes’ law 192 stopping sight distance 162 strain 33 Streeter Phelps 175 stress 33 stress, pressure, and viscosity 62 stress-strain curve for mild steel 33 stress and strain 34 stress concentration in brittle materials 105 stresses in beams 37 stresses in spur gears 238 structural analysis 142 structural design 143 structure of matter 104 student’s t-distribution table 46 subdivisions of biological organisms 91 submerged orifice 68 Superpave 167 surface tension 62 surface tension and capillarity 62 sweep-through concentration change in a vessel 133 switching function terminology 214 systems of forces 49

T t-distribution 42 t-distribution table 46 taxation 115 Taylor’s series 25 Taylor tool life formula 223 temperature-entropy (T-s) diagram 77 temperature coefficient of expansion 33 temperature conversions 19 tension members 151 test test for a maximum 25 test for a minimum 25 test for a point of inflection 25 tests for out-of-control 216

thermal and mechanical processing 104 thermal deformations 33 thermal resistance 84 thermo-mechanical properties of polymers 107 thermodynamics 73 chemical thermodynamics 125 thermodynamic cycles 78 thermodynamic system 74 thermodynamic system 74 threaded fasteners 233 three-phase systems 197 threshold limit value (TLV) 132 time constant 112 torsion 36 torsional strain 36 torsional vibration 60 torsion springs 231 toxicology 180 dose-response curves 180 selected chemical interaction effects 180 traffic flow relationships 163 transducer sensitivity 110 transformers (ideal) 198 transient conduction using the lumped capacitance method 85 transmission lines 199 transportation 162 transport phenomena 90 trapezoidal rule 32, 165 trigonometry 22 truss 50 truss deflection by unit load method 142 turbines 243, 246 Turns ratio 198 two-film theory (for equimolar counter-diffusion) 128 two-port parameters 197 two-way ANOVA table 224

U ultrafiltration 192 undamped natural frequency 60 uniaxial loading and deformation 33 uniaxial stress-strain 33 unit normal distribution table 45 units 19 electrical 193 universal gas constant 189, 192 unsaturated acyclic hydrocarbons 123

V vapor-liquid equilibrium 125 vapor-liquid mixtures 76 variable loading failure theories 35 variance for 2n factorial designs 219 vectors 24 vehicle signal change interval 162 velocity gradient 191 venturi meters 68 vertical curve formulas 165 vertical stress beneath a uniformly-loaded circular area 139

255

vertical stress caused by a point load 139 vibrations 236 viscosity 62 voltage 193 volume formulas 28 volumetric efficiency 242 von Mises stress 232

W wastewater operating parameters 186 wastewater treatment and technologies 186 water-cement ratio 106 water treatment technologies 188 Weir formulas 161 well drawdown 159 wet-bulb temperature 76 work 56, 57 work sampling formulas 223

X Y Z z-transforms 201 zero-order reaction 126

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